The Derivative of e^-x

November 9, 2020October 18, 2021 ~ DerivativeIt

The derivative of e^-x is -e^-x

How to calculate the derivative of e^-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 e^x (the answer is e^x)
We know how to differentiate -x (the answer is -1)

Because e^-x is a function which is a combination of e^x and -x, it means we can perform the differentiation of e to the -x by making use of the chain rule.

Using the chain rule to find the derivative of e^-x

Although the function e^-x contains no parenthesis, we can still view it as a composite function (a function of a function).

If we add parenthesis around the exponent, we get e^(-x).

Now the function is in the form of the standard exponential function e^x, except it does not have x as an exponent, instead the exponent is another function of x (-x).

Let’s call the function in the exponent g(x), which means:

g(x) = -x

From this it follows that:

e^-x = e^g(x)

Let’s set f(x) = e^x.

Then, because g(x) = -x, the function e^-x can be written as a composite function of f(x) and g(x).

f(x) = e^x

f(g(x)) = e^g(x) (but g(x) = -x)

Therefore, f(g(x)) = e^-x

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

F(x) = f(g(x)) = e^-x

We can now find the derivative of F(x) = e^-x, F'(x), by making use of the chain rule.

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)

Now we can just plug f(x) and g(x) into the chain rule to find the derivative of e to the negative x.

How to find the derivative of e^-x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(-1)	g(x) = -x ⇒ g'(x) = –1
	= (e^-x).(-1)	f(g(x)) = e^-x ⇒ f'(g(x)) = e^-x
	= -e^(-x)

Using the chain rule, the derivative of e^-x is -e^-x

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

e^-x	► Derivative of e^-x = -e^-x
e^(-x)	► Derivative of e^(-x) = -e^-x
e -x	► Derivative of e -x = -e^-x
e – x	► Derivative of e – x = -e^-x
e to the negative x	► Derivative of e to the negative x = -e^-x
e to the -x	► Derivative of e to the -x = -e^-x

Top Tip

It’s possible to generalize the derivative of expressions in the form e^ax (where a is a constant value):

The derivative of e^ax = ae^ax

(Add the constant a to the front of the expression and keep the exponential part the same)

The Second Derivative of e^-x

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

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

We can use the chain rule to calculate the derivative of -e^-x and get an answer of e^-x (i.e. the second derivative of e^-x is just itself).

► The second derivative of e^-x = e^(-x)

The Derivative of cos(3x)

November 9, 2020October 18, 2021 ~ DerivativeIt

The derivative of cos(3x) is -3sin(3x)

How to calculate the derivative of cos(3x)

Note that in this post we will be looking at differentiating cos(3x) which is not the same as differentiating cos³x). Here is our post dealing with how to differentiate cos³(x).

In this case:

We know how to differentiate cos(x) (the answer is -sin(x))
We know how to differentiate 3x (the answer is 3)

This means the chain rule will allow us to differentiate the expression cos(3x).

Using the chain rule to find the derivative of cos(3x)

To perform the differentiation cos(3x), 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 3x).

Let’s call the function in the argument of cos, g(x), which means the function is in the form of cos(x), except it does not have x as the angle, instead it has another function of x (3x) as the angle

If:

g(x) = 3x

It follows that:

cos(3x) = cos(g(x))

So if the function f(x) = cos(x) and the function g(x) = 3x, then the function cos(3x) can be written as a composite function.

f(x) = cos(x)

f(g(x)) = cos(g(x)) (but g(x) = 3x))

f(g(x)) = cos(3x)

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

F(x) = f(g(x)) = cos(3x)

We can find the derivative of cos(3x) (F'(x)) by making use of the chain rule.

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)

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 cos function.

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

In a similar way, the derivative of cos(3x) with respect to 3x is -sin(3x).

We will use this fact as part of the chain rule to find the derivative of cos(3x) with respect to x.

How to find the derivative of cos(3x) using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(3)	g(x) = 3x ⇒ g‘(x) = 3
	= (-sin(3x)).(3)	f(g(x)) = cos(3x) ⇒ f'(g(x)) = -sin(3x)
	= -3sin(3x)

Using the chain rule, the derivative of cos(3x) is -3sin(3x)

Finally, just a note on syntax and notation: cos(3x) 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.

cos3x	► Derivative of cos3x = -3sin(3x)
cos 3 x	► Derivative of cos 3 x = -3sin(3x)
cos 3x	► Derivative of cos 3x = -3sin(3x)
cos (3x)	► Derivative of cos (3x) = -3sin(3x)

The Second Derivative Of cos(3x)

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

From above, we found that the first derivative of cos(3x) = -3sin(3x). So to find the second derivative of cos(3x), we just need to differentiate -3sin(3x)

We can use the chain rule to find the derivative of -3sin(3x) and it gives us a result of -9cos(3x)

► The second derivative of cos(3x) is -9cos(3x)

The Derivative of e^-2x

November 9, 2020 ~ DerivativeIt

The derivative of e^-2x is -2e^-2x

How to calculate the derivative of e^-2x

In this case:

We know how to differentiate e^x (the answer is e^x)
We know how to differentiate -2x (the answer is -2)

Because e^-2x is a function which is a combination of e^x and -2x, it means we can perform the differentiation of e to the -2x by making use of the chain rule.

Using the chain rule to find the derivative of e^-2x

Although the function e^-2x contains no parenthesis, we can still view it as a composite function (a function of a function).

If we add parenthesis around the exponent, we get e^(-2x).

Now the function is in the form of the standard exponential function e^x, except it does not have x as an exponent, instead the exponent is another function of x (-2x).

Let’s call the function in the exponent g(x), which means:

g(x) = -2x

From this it follows that:

e^-2x = e^g(x)

Let’s set f(x) = e^x.

Then, because g(x) = -2x, the function e^-2x can be written as a composite function of f(x) and g(x).

f(x) = e^x

f(g(x)) = e^g(x) (but g(x) = -2x)

Therefore, f(g(x)) = e^-2x

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

F(x) = f(g(x)) = e^-2x

We can now find the derivative of F(x) = e^-2x, F'(x), by making use of the chain rule.

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)

Now we can just plug f(x) and g(x) into the chain rule to find the derivative of e to the -2x.

How to find the derivative of e^-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
	= (e^-2x).(-2)	f(g(x)) = e^-2x ⇒ f'(g(x)) = e^-2x
	= -2e^(-2x)

Using the chain rule, the derivative of e^-2x is -2e^-2x

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

e^-2x	► Derivative of e^-2x = -2e^-2x
e^(-2x)	► Derivative of e^(-2x) = -2e^-2x
e -2x	► Derivative of e -2x = -2e^-2x
e -2 x	► Derivative of e -2 x = -2e^-2x
e to the -2x	► Derivative of e to the -2x = -2e^-2x

The Second Derivative of e^-2x

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

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

We can use the chain rule to calculate the derivative of -2e^-2x and get an answer of 4e^-2x.

► The second derivative of e^-2x = 4e^(-2x)

The Derivative of e^-3x

November 9, 2020 ~ DerivativeIt

The derivative of e^-3x is -3e^-3x

How to calculate the derivative of e^-3x

In this case:

We know how to differentiate e^x (the answer is e^x)
We know how to differentiate -3x (the answer is -3)

Because e^-3x is a function which is a combination of e^x and -3x, it means we can perform the differentiation of e to the -3x by making use of the chain rule.

Using the chain rule to find the derivative of e^-3x

Although the function e^-3x contains no parenthesis, we can still view it as a composite function (a function of a function).

If we add parenthesis around the exponent, we get e^(-3x).

Now the function is in the form of the standard exponential function e^x, except it does not have x as an exponent, instead the exponent is another function of x (-3x).

Let’s call the function in the exponent g(x), which means:

g(x) = -3x

From this it follows that:

e^-3x = e^g(x)

Let’s set f(x) = e^x.

Then, because g(x) = -3x, the function e^-3x can be written as a composite function of f(x) and g(x).

f(x) = e^x

f(g(x)) = e^g(x) (but g(x) = -3x)

Therefore, f(g(x)) = e^-3x

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

F(x) = f(g(x)) = e^-3x

We can now find the derivative of F(x) = e^-3x, F'(x), by making use of the chain rule.

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)

Now we can just plug f(x) and g(x) into the chain rule to find the derivative of e to the -3x.

How to find the derivative of e^-3x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(-3)	g(x) = -3x ⇒ g'(x) = -3
	= (e^-3x)(-3)	f(g(x)) = e^-3x ⇒ f'(g(x)) = e^-3x
	= -3e^(-3x)

Using the chain rule, the derivative of e^-3x is -3e^-3x

Finally, just a note on syntax and notation: the exponential function e^-3x is sometimes written in the forms shown below (the derivative of each is as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

e^-3x	► Derivative of e^-3x = -3e^-3x
e^(-3x)	► Derivative of e^(-3x) = -3e^-3x
e -3x	► Derivative of e -3x = -3e^-3x
e -3 x	► Derivative of e -3 x = -3e^-3x
e to the -3x	► Derivative of e to the -3x = -3e^-3x

The Second Derivative of e^-3x

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

From above, we found that the first derivative of e^-3x = -3e^(-3x). So to find the second derivative of e^-3x, we just need to differentiate -3e^-3x

We can use the chain rule to calculate the derivative of -3e^-3x and get an answer of 9e^-3x.

► The second derivative of e^-3x = 9e^(-3x)