The Derivative of e^5x

September 30, 2020 ~ DerivativeIt

The derivative of e^5x is 5e^5x

How to calculate the derivative of e^5x

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 5x (the answer is 5)

Because e^5x is a function which is a combination of e^x and 5x, it means we can perform the differentiation of e to the 5x by making use of the chain rule.

Using the chain rule to find the derivative of e^5x

Although the function e^5x 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^(5x).

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 (5x).

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

g(x) = 5x

From this it follows that:

e^5x = e^g(x)

Let’s set f(x) = e^x.

Then, because g(x) = 5x, the function e^5x 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) = 5x)

Therefore, f(g(x)) = e^5x

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

F(x) = f(g(x)) = e^5x

We can now find the derivative of F(x) = e^5x, 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 5x.

How to find the derivative of e^5x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(5)	g(x) = 5x ⇒ g'(x) = 5
	= (e^5x).5	f(g(x)) = e^5x ⇒ f'(g(x)) = e^5x
	= 5e^(5x)

Using the chain rule, the derivative of e^5x is 5e^5x

Finally, just a note on syntax and notation: the exponential function e^5x 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^5x	► Derivative of e^5x = 5e^5x
e^(5x)	► Derivative of e^(5x) = 5e^5x
e 5x	► Derivative of e 5x = 5e^5x
e 5 x	► Derivative of e 5 x = 5e^5x
e to the 5x	► Derivative of e to the 5x = 5e^5x

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^5x

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

From above, we found that the first derivative of e^5x = 5e^(5x). So to find the second derivative of e^5x, we just need to differentiate 5e^5x

We can use the chain rule to calculate the derivative of 5e^5x and get an answer of 25e^5x.

► The second derivative of e^5x = 25e^(5x)

The Derivative of e^x^2

September 28, 2020 ~ DerivativeIt

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

How to calculate the derivative of e^x^2

In this case:

We know how to differentiate e^x (the answer is e^x)
We know how to differentiate x² (the answer is 2x)

Because e^x^2 is a function which is a combination of e^x and x², it means we can perform the differentiation of e to the x² by making use of the chain rule.

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

Although the function e^x² 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^(x²).

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 (x²).

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

g(x) = x²

From this it follows that:

e^x² = e^g(x)

Let’s set f(x) = e^x.

Then, because g(x) = x², the function e^x² 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) = x²)

Therefore, f(g(x)) = e^x²

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

F(x) = f(g(x)) = e^x²

We can now find the derivative of F(x) = e^x^2, 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 x squared.

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

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(2x)	g(x) = x² ⇒ g'(x) = 2x
	= (e^x^2).(2x)	f(g(x)) = e^x^2 ⇒ f'(g(x)) = e^x^2
	= 2xe^x^2

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

Finally, just a note on syntax and notation: the exponential function e^x^2 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^x²	► Derivative of e^x² = 2xe^x²
e^(x^2)	► Derivative of e^(x^2) = 2xe^x²
e x 2	► Derivative of e x 2 = 2xe^x²
e to the x squared	► Derivative of e to the x squared = 2xe^x²

The Second Derivative of e^x^2

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^2 = 2xe^x². So to find the second derivative of e^x^2, we just need to differentiate 2xe^x²

We can use the chain rule in combination with the product rule for differentiation to calculate the derivative of 2xe^x² and get an answer of 4x²e^x² + 2e^x²

► The second derivative of e^x^2 = 4x²e^x² + 2e^x²

The Derivative of e^4x

September 25, 2020 ~ DerivativeIt

The derivative of e^4x is 4e^4x

How to calculate the derivative of e^4x

In this case:

We know how to differentiate e^x (the answer is e^x)
We know how to differentiate 4x (the answer is 4)

Because e^4x is a function which is a combination of e^x and 4x, it means we can perform the differentiation of e to the 4x by making use of the chain rule.

Using the chain rule to find the derivative of e^4x

Although the function e^4x 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^(4x).

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 (4x).

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

g(x) = 4x

From this it follows that:

e^4x = e^g(x)

Let’s set f(x) = e^x.

Then, because g(x) = 4x, the function e^4x 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) = 4x)

Therefore, f(g(x)) = e^4x

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

F(x) = f(g(x)) = e^4x

We can now find the derivative of F(x) = e^4x, 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 4x.

How to find the derivative of e^4x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(4)	g(x) = 4x ⇒ g'(x) = 4
	= (e^4x).4	f(g(x)) = e^4x ⇒ f'(g(x)) = e^4x
	= 4e^(4x)

Using the chain rule, the derivative of e^4x is 4e^4x

Finally, just a note on syntax and notation: the exponential function e^4x 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^4x	► Derivative of e^4x = 4e^4x
e^(4x)	► Derivative of e^(4x) = 4e^4x
e 4x	► Derivative of e 4x = 4e^4x
e 4 x	► Derivative of e 4 x = 4e^4x
e to the 4x	► Derivative of e to the 4x = 4e^4x

The Second Derivative of e^4x

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

From above, we found that the first derivative of e^4x = 4e^(4x). So to find the second derivative of e^4x, we just need to differentiate 4e^4x

We can use the chain rule to calculate the derivative of 4e^4x and get an answer of 16e^4x.

► The second derivative of e^4x = 16e^(4x)

The Derivative of e^7x

September 14, 2020 ~ DerivativeIt

The derivative of e^7x is 7e^7x

How to calculate the derivative of e^7x

In this case:

We know how to differentiate e^x (the answer is e^x)
We know how to differentiate 7x (the answer is 7)

Because e^7x is a function which is a combination of e^x and 7x, it means we can perform the differentiation of e to the 7x by making use of the chain rule.

Using the chain rule to find the derivative of e^7x

Although the function e^7x 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^(7x).

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 (7x).

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

g(x) = 7x

From this it follows that:

e^7x = e^g(x)

Let’s set f(x) = e^x.

Then, because g(x) = 7x, the function e^7x 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) = 7x)

Therefore, f(g(x)) = e^7x

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

F(x) = f(g(x)) = e^7x

We can now find the derivative of F(x) = e^7x, 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 7x.

How to find the derivative of e^7x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(7)	g(x) = 7x ⇒ g'(x) = 7
	= (e^7x).7	f(g(x)) = e^7x ⇒ f'(g(x)) = e^7x
	= 7e^(7x)

Using the chain rule, the derivative of e^7x is 7e^7x

Finally, just a note on syntax and notation: the exponential function e^7x 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^7x	► Derivative of e^7x = 7e^7x
e^(7x)	► Derivative of e^(7x) = 7e^7x
e 7x	► Derivative of e 7x = 7e^7x
e 7 x	► Derivative of e 7 x = 7e^7x
e to the 7x	► Derivative of e to the 7x = 7e^7x

The Second Derivative of e^7x

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

From above, we found that the first derivative of e^7x = 7e^(7x). So to find the second derivative of e^7x, we just need to differentiate 7e^7x

We can use the chain rule to calculate the derivative of 7e^7x and get an answer of 49e^7x.

► The second derivative of e^7x = 49e^(7x)

The Derivative of e^2x

September 11, 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

August 5, 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)