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The derivative of sin(2x) is 2cos(2x)

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How to calculate the derivative of sin(2x)

Note that in this post we will be looking at differentiating sin(2x) which is not the same as differentiating sin²(x). Here is our post dealing with how to differentiate sin^2(x).

The chain rule is useful for finding the derivative of a function which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate sin(x) (the answer is cos(x))
• We know how to differentiate 2x (the answer is 2)

This means the chain rule will allow us to differentiate the expression sin(2x).

Using the chain rule to find the derivative of sin(2x)

To perform the differentiation sin(2x), the chain rule says we must differentiate the expression as if it were just in terms of x as long as we then multiply that result by the derivative of what the expression is actually in terms of (in this case the derivative of 2x).

Let’s call the function in the argument of sin, g(x), which means the function is in the form of sin(x), except it does not have x as the angle, instead it has another function of x (2x) as the angle.

If:

g(x) = 2x

It follows that:

sin(2x) = sin(g(x))

So if the function f(x) = sin(x) and the function g(x) = 2x, then the function sin(2x) can be written as a composite function.

f(x) = sin(x)

f(g(x)) = sin(g(x)) (but g(x) = 2x))

f(g(x)) = sin(2x)

Let’s define this composite function as F(x):

F(x) = f(g(x)) = sin(2x)

We can find the derivative of sin(2x) (F'(x)) by making use of the chain rule.

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The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

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Now we can just plug f(x) and g(x) into the chain rule. But before we do that, just a quick recap on the derivative of the sin function.

The derivative of sin(x) with respect to x is cos(x)
The derivative of sin(z) with respect to z is cos(z)

In a similar way, the derivative of sin(2x) with respect to 2x is cos(2x).

We will use this fact as part of the chain rule to find the derivative of sin(2x) with respect to x.

How to find the derivative of sin(2x) using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(2)	g(x) = 2x ⇒ g'(x) = 2
	= (cos(2x)).(2)	f(g(x)) = sin(2x) ⇒ f'(g(x)) = cos(2x)
	= 2cos(2x)

Using the chain rule, the derivative of sin(2x) is 2cos(2x)

Interestingly, the first derivative of sin(2x) is equal to the second derivative of sin²(x). The reason for this is because the derivative of sin²(x) is equal to sin(2x), which is what we just differentiated.

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Finally, just a note on syntax and notation: sin(2x) is sometimes written in the forms below (with the derivative as per the calculation above). Just be aware that not all of the forms below are mathematically correct.

sin2x	► Derivative of sin2x = 2cos(2x)
sin 2 x	► Derivative of sin 2 x = 2cos(2x)
sin 2x	► Derivative of sin 2x = 2cos(2x)
sin (2x)	► Derivative of sin (2x) = 2cos(2x)

The Second Derivative Of sin(2x)

To calculate the second derivative of a function, you just differentiate the first derivative.

From above, we found that the first derivative of sin(2x) = 2cos(2x). So to find the second derivative of sin(2x), we just need to differentiate 2cos(2x)

We can use the chain rule to find the derivative of 2cos(2x) and it gives us a result of -4sin(2x)

► The second derivative of sin(2x) is -4sin(2x)

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The derivative of cot^2x is -2.csc^2(x).cot(x)

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How to calculate the derivative of cot^2x

There are two methods that can be used for calculating the derivative of cot^2x.

The first method is by using the product rule for derivatives (since cot²(x) can be written as cot(x).cot(x)).

The second method is by using the chain rule for differentiation.

Finding the derivative of cot^2x using the product rule

The product rule for differentiation states that the derivative of f(x).g(x) is f’(x)g(x) + f(x).g’(x)

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The Product Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(x).g(x)

Then the derivative of F(x) is F'(x) = f’(x)g(x) + f(x)g'(x)

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First, let F(x) = cot²(x)

Then remember that cot²(x) is equal to cot(x).cot(x)

So F(x) = cot(x)cot(x)

By setting f(x) and g(x) as cot(x) means that F(x) = f(x).g(x) and we can apply the product rule to find F'(x)

F'(x)	= f'(x)g(x) + f(x)g'(x)	Product Rule Definition
	= f'(x)cot(x) + cot(x)g'(x)	f(x) = g(x) = cot(x)
	= -csc2(x)cot(x) + cot(x)(-csc2(x))	f'(x) = g(‘x) = -csc2(x)
	= -2csc2(x)cot(x)

Using the product rule, the derivative of cot^2x is -2csc²(x)cot(x)

Finding the derivative of cot^2x using the chain rule

The chain rule is useful for finding the derivative of a function which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate cot(x) (the answer is -csc²(x))
• We know how to differentiate x² (the answer is 2x)

This means the chain rule will allow us to perform the differentiation of the expression cot^2x.

Using the chain rule to find the derivative of cot^2x

Although the expression cot²x contains no parenthesis, we can still view it as a composite function (a function of a function).

We can write cot²x as (cot(x))².

Now the function is in the form of x², except it does not have x as the base, instead it has another function of x (cot(x)) as the base.

Let’s call the function of the base g(x), which means:

g(x) = cot(x)

From this it follows that:

(cot(x))² = g(x)²

So if the function f(x) = x² and the function g(x) = cot(x), then the function (cot(x))² can be written as a composite function.

f(x) = x²

f(g(x)) = g(x)² (but g(x) = cot(x))

f(g(x)) = (cot(x))²

Let’s define this composite function as F(x):

F(x) = f(g(x)) = (cot(x))²

We can find the derivative of cot^2x (F'(x)) by making use of the chain rule.

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The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

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Now we can just plug f(x) and g(x) into the chain rule.

How to find the derivative of cot^2x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(-csc2(x))	g(x) = cot(x) ⇒ g'(x) = -csc2(x)
	= (2.cot(x)).(-csc2(x))	f(g(x)) = (cot(x))2 ⇒ f'(g(x)) = 2.cot(x)
	= -2csc2(x)cot(x)

Using the chain rule, the derivative of cot^2x is -2.csc²(x)cot(x)

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Finally, just a note on syntax and notation:cot^2x is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

cot2x	► Derivative of cot2x = -2.csc2(x).cot(x)
cot^2(x)	► Derivative of cot^2(x) = -2.csc2(x).cot(x)
cot 2 x	► Derivative of cot 2 x = -2.csc2(x).cot(x)
(cotx)^2	► Derivative of (cotx)^2 = -2.csc2(x).cot(x)
cot squared x	► Derivative of cot squared x = -2.csc2(x).cot(x)
cotx2	► Derivative of cotx2 = -2.csc2(x).cot(x)
cot^2	► Derivative of cot^2 = -2.csc2(x).cot(x)

The Second Derivative Of cot^2x

To calculate the second derivative of a function, differentiate the first derivative.

From above, we found that the first derivative of cot^2x = -2csc²(x)cot(x). So to find the second derivative of cot^2x, we need to differentiate -2csc²(x)cot(x).

We can use the product and chain rules, and then simplify to find the derivative of -2csc²(x)cot(x) is 4csc²(x)cot²(x) + 2csc⁴(x)

► The second derivative of cot^2x is 4csc²(x)cot²(x) + 2csc⁴(x)

Interesting property of the derivative of cot^2x

It is interesting to note that the derivative of cot²x is equal to the derivative of csc²x.

The derivative of:
> cot²x = -2.csc²(x).cot(x)
> csc²x = -2.csc²(x).cot(x)

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The derivative of csc^2x is -2.csc^2(x).cot(x)

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How to calculate the derivative of csc^2x

Note that in this post we will be looking at differentiating csc²(x) which is not the same as differentiating csc(2x). Here is our post dealing with how to differentiate csc(2x).

There are two methods that can be used for calculating the derivative of cosec^2x.

The first method is by using the product rule for derivatives (since csc²(x) can be written as csc(x).csc(x)).

The second method is by using the chain rule for differentiation.

Finding the derivative of csc^2x using the product rule

The product rule for differentiation states that the derivative of f(x).g(x) is f’(x)g(x) + f(x).g’(x)

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The Product Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(x).g(x)

Then the derivative of F(x) is F'(x) = f’(x)g(x) + f(x)g'(x)

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First, let F(x) = csc²(x)

Then remember that csc²(x) is equal to csc(x).csc(x)

So F(x) = csc(x)csc(x)

By setting f(x) and g(x) as csc(x) means that F(x) = f(x).g(x) and we can apply the product rule to find F'(x)

F'(x)	= f'(x)g(x) + f(x)g'(x)	Product Rule Definition
	= f'(x)csc(x) + csc(x)g'(x)	f(x) = g(x) = csc(x)
	= -csc(x)cot(x)csc(x) + csc(x)(-csc(x)cot(x))	f'(x) = g(‘x) = -csc(x)cot(x)
	= -2csc2(x)cot(x)

Using the product rule, the derivative of csc^2x is -2csc^2(x)cot(x)

Finding the derivative of csc^2x using the chain rule

The chain rule is useful for finding the derivative of a function which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate csc(x) (the answer is -csc(x)cot(x))
• We know how to differentiate x² (the answer is 2x)

This means the chain rule will allow us to perform the differentiation of the expression csc^2x.

Using the chain rule to find the derivative of csc^2x

Although the expression csc²x contains no parenthesis, we can still view it as a composite function (a function of a function).

We can write csc²x as (csc(x))².

Now the function is in the form of x², except it does not have x as the base, instead it has another function of x (csc(x)) as the base.

Let’s call the function of the base g(x), which means:

g(x) = csc(x)

From this it follows that:

csc(x)² = g(x)²

So if the function f(x) = x² and the function g(x) = csc(x), then the function (csc(x))² can be written as a composite function.

f(x) = x²

f(g(x)) = g(x)² (but g(x) = csc(x))

f(g(x)) = (csc(x))²

Let’s define this composite function as F(x):

F(x) = f(g(x)) = (csc(x))²

We can find the derivative of csc^2x (F'(x)) by making use of the chain rule.

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The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

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Now we can just plug f(x) and g(x) into the chain rule.

How to find the derivative of csc^2x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(-csc(x)cot(x))	g(x) = csc(x) ⇒ g'(x) = -csc(x)cot(x)
	= (2csc(x)).(csc(x)tan(x))	f(g(x)) = (csc(x))2 ⇒ f'(g(x)) = 2csc(x)
	= -2csc2(x)cot(x)

Using the chain rule, the derivative of csc^2x is -2.csc^2(x).cot(x)

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Finally, just a note on syntax and notation: csc^2x is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

csc2x	► Derivative of csc2x = -2csc2(x)cot(x)
csc^2(x)	► Derivative of csc^2(x) = -2csc2(x)cot(x)
csc 2 x	► Derivative of csc 2 x = -2csc2(x)cot(x)
(cscx)^2	► Derivative of (cscx)^2 = -2csc2(x)cot(x)
csc squared x	► Derivative of csc squared x = -2csc2(x)cot(x)
cscx2	► Derivative of cscx2 = -2csc2(x)cot(x)
csc^2	► Derivative of csc^2 = -2csc2(x)cot(x)

The Second Derivative Of csc^2x

To calculate the second derivative of a function, differentiate the first derivative.

From above, we found that the first derivative of csc^2x = -2csc²(x)cot(x). So to find the second derivative of csc^2x, we need to differentiate -2csc²(x)cot(x).

We can use the product and chain rules, and then simplify to find the derivative of -2csc²(x)cot(x) is 4csc²(x)cot²(x) + 2csc⁴(x)

► The second derivative of csc^2x is 4csc²(x)cot²(x) + 2csc⁴(x)

Interesting property of the derivative of csc^2x

It is interesting to note that the derivative of csc²x is equal to the derivative of cot²x.

The derivative of:
> csc²x = -2.csc²(x).cot(x)
> cot²x = -2.csc²(x).cot(x)

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The derivative of sec^2x is 2.sec^2(x).tan(x)

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How to calculate the derivative of sec^2x

There are two methods that can be used for calculating the derivative of sec^2x.

The first method is by using the product rule for derivatives (since sec²(x) can be written as sec(x).sec(x)).

The second method is by using the chain rule for differentiation.

Finding the derivative of sec^2x using the product rule

The product rule for differentiation states that the derivative of f(x).g(x) is f’(x)g(x) + f(x).g’(x)

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The Product Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(x).g(x)

Then the derivative of F(x) is F'(x) = f’(x)g(x) + f(x)g'(x)

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First, let F(x) = sec²(x)

Then remember that sec²(x) is equal to sec(x).sec(x)

So F(x) = sec(x)sec(x)

By setting f(x) and g(x) as sec(x) means that F(x) = f(x).g(x) and we can apply the product rule to find F'(x)

F'(x)	= f'(x)g(x) + f(x)g'(x)	Product Rule Definition
	= f'(x)sec(x) + sec(x)g'(x)	f(x) = g(x) = sec(x)
	= sec(x)tan(x)sec(x) + sec(x)sec(x)tan(x)	f'(x) = g(‘x) = sec(x)tan(x)
	= 2sec2(x)tan(x)

Using the product rule, the derivative of sec^2x is 2sec^2(x)tan(x)

Finding the derivative of sec^2x using the chain rule

The chain rule is useful for finding the derivative of a function which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate sec(x) (the answer is sec(x)tan(x))
• We know how to differentiate x² (the answer is 2x)

This means the chain rule will allow us to perform the differentiation of the expression sec^2x.

Using the chain rule to find the derivative of sec^2x

Although the expression sec²x contains no parenthesis, we can still view it as a composite function (a function of a function).

We can write sec²x as (sec(x))².

Now the function is in the form of x², except it does not have x as the base, instead it has another function of x (sec(x)) as the base.

Let’s call the function of the base g(x), which means:

g(x) = sec(x)

From this it follows that:

sec(x)² = g(x)²

So if the function f(x) = x² and the function g(x) = sec(x), then the function (sec(x))² can be written as a composite function.

f(x) = x²

f(g(x)) = g(x)² (but g(x) = sec(x))

f(g(x)) = (sec(x))²

Let’s define this composite function as F(x):

F(x) = f(g(x)) = (sec(x))²

We can find the derivative of sec^2x (F'(x)) by making use of the chain rule.

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The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

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Now we can just plug f(x) and g(x) into the chain rule.

How to find the derivative of sec^2x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(sec(x)tan(x))	g(x) = sec(x) ⇒ g'(x) = sec(x)tan(x)
	= (2sec(x)).(sec(x)tan(x))	f(g(x)) = (sec(x))2 ⇒ f'(g(x)) = 2sec(x)
	= 2sec2(x)tan(x)

Using the chain rule, the derivative of sec^2x is 2.sec^2(x).tan(x)

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Finally, just a note on syntax and notation: sec^2x is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

sec2x	► Derivative of sec2x = 2sec2(x)tan(x)
sec^2(x)	► Derivative of sec^2(x) =2sec2(x)tan(x)
sec 2 x	► Derivative of sec 2 x = 2sec2(x)tan(x)
(secx)^2	► Derivative of (secx)^2 =2sec2(x)tan(x)
sec squared x	► Derivative of sec squared x = 2sec2(x)tan(x)
secx2	► Derivative of secx2 = 2sec2(x)tan(x)
sec^2	► Derivative of sec^2 = 2sec2(x)tan(x)

The Second Derivative Of sec^2x

To calculate the second derivative of a function, differentiate the first derivative.

From above, we found that the first derivative of sec^2x = 2sec²(x)tan(x). So to find the second derivative of sec^2x, we need to differentiate 2sec²(x)tan(x).

We can use the product and chain rules, and then simplify to find the derivative of 2sec²(x)tan(x) is 4sec²(x)tan²(x) + 2sec⁴(x)

► The second derivative of sec^2x is 4sec²(x)tan²(x) + 2sec⁴(x)

Interesting property of the derivative of sec^2x

It is interesting to note that the derivative of sec²(x) is equal to the derivative of tan²(x).

The derivative of:
> sec²x = 2.sec²(x).tan(x)
> tan²x = 2.sec²(x).tan(x)