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iGCSE (2021 Edition)

11.13 Differentiating exponential functions

Lesson

We saw previously that we can differentiate the exponential function as follows: 

Standard forms for differentiating exponential functions
$\frac{d}{dx}e^x$ddxex $=$= $e^x$ex
$\frac{d}{dx}e^{ax}$ddxeax $=$= $ae^{ax}$aeax

We will now look at how to differentiate more general forms of the exponential function.

Differentiating $y=e^{f\left(x\right)}$y=ef(x)

Consider functions of the form:

$y$y $=$= $e^{f\left(x\right)}$ef(x)
 

Letting $u=f\left(x\right)$u=f(x), we have:

$y$y $=$= $e^u$eu
$\frac{dy}{du}$dydu $=$= $e^u$eu
 

And:

$\frac{du}{dx}$dudx $=$= $f'\left(x\right)$f(x)
 

Now using the chain rule, we find that:

$\frac{dy}{dx}$dydx $=$= $\frac{dy}{du}\times\frac{du}{dx}$dydu×dudx
$\frac{dy}{dx}$dydx $=$= $e^u\times u'$eu×u
$\frac{dy}{dx}$dydx $=$= $f'\left(x\right)e^{f\left(x\right)}$f(x)ef(x)
 
Differentiating $y=e^{f\left(x\right)}$y=ef(x)
$\frac{d}{dx}e^{f\left(x\right)}$ddxef(x) $=$= $f'\left(x\right)e^{f\left(x\right)}$f(x)ef(x)

Worked examples

Example 1

Differentiate the function $y=e^{3x^2-x-2}$y=e3x2x2.

Solution: Using the rule that we developed above, we have that:

$\frac{dy}{dx}$dydx $=$= $\left(6x-1\right)e^{3x^2-x-2}$(6x1)e3x2x2
Example 2

Differentiate the function $y=\frac{e^{2x}-e^{-x}}{e^x}$y=e2xexex.

Think: This function has exponentials of the same base in its numerator and denominator. So it will be easier to differentiate if we first rewrite the function by splitting the fraction and simplifying the exponentials.

Do: Rewriting the function first:

$y$y $=$= $\frac{e^{2x}-e^{-x}}{e^x}$e2xexex
$y$y $=$= $\frac{e^{2x}}{e^x}-\frac{e^{-x}}{e^x}$e2xexexex
$y$y $=$= $e^x-e^{-2x}$exe2x

 

Now we can more easily differentiate:

$\frac{dy}{dx}$dydx $=$= $e^x+2e^{-2x}$ex+2e2x

 

Practice questions

Question 1

Differentiate $y=\left(2x-e^{3x}\right)^7$y=(2xe3x)7.

Question 2

Find the derivative of $y=x^5e^{4x}$y=x5e4x. Express the derivative in factorised form.

(Note: You may let $u=x^5$u=x5)

Question 3

Differentiate $y=\frac{e^{6x}}{1+e^x}$y=e6x1+ex.

Differentiating $y=a^x$y=ax

In calculus, there are functions that first need to be manipulated into another form in order for them to be differentiated. The function $y=a^x$y=ax fits this criterion.

We know that the exponential function is a special case. When $y=e^x$y=ex is differentiated it gives $y'=e^x$y=ex. That is, when the base is $e\approx2.7182818...$e2.7182818...:

$y=\frac{dy}{dx}=e^x$y=dydx=ex

Consider the applet below which shows the function $y=a^x$y=ax and its derivative (dotted line). Notice that the derivative of any exponential function, where $a\ne e$ae, is another similar exponential function.

 

From the applet, we could say that the derivative of $y=a^x$y=ax, is the original function multiplied by some constant.  That is:

$\frac{dy}{dx}=ka^x$dydx=kax, where $k$k is a constant.

We will use logarithm laws to determine this constant and find the derivative of $y=a^x$y=ax for the case when $a\ne e$ae.

$y$y $=$= $a^x$ax

 

Taking the natural log of both sides gives:

$\ln y$lny $=$= $\ln a^x$lnax

 

Using our log laws gives:

$\ln y$lny $=$= $x\ln a$xlna

 

Reversing the process, we get:

$e^{\ln y}$elny $=$= $e^{x\ln a}$exlna

 

Which can be simplified to:

$y$y $=$= $e^{x\ln a}$exlna
 

Taking the derivative of both sides using $\frac{d}{dx}e^{f(x)}=f'\left(x\right)e^{f\left(x\right)}$ddxef(x)=f(x)ef(x):

$\frac{dy}{dx}$dydx $=$= $\ln ae^{x\ln a}$lnaexlna
 

But, remember $y=e^{x\ln a}=a^x$y=exlna=ax, therefore:

$\frac{dy}{dx}$dydx $=$= $\ln a\ a^x$lna ax
 

Therefore the constant term is the natural log of the base number. This agrees with what we know for the exponential function $y=e^x$y=ex, as $\ln e=1$lne=1.

Differentiating $y=a^x$y=ax
$\frac{d}{dx}a^x$ddxax $=$= $\ln a\ a^x$lna ax

Worked example

Example 3

Differentiate the function $y=2^x$y=2x.

Think:  This is an exponential function with $a\ne e$ae. Therefore we will use $\frac{d}{dx}a^x=\ln a\ a^x$ddxax=lna ax.

Do:  

$y$y $=$= $2^x$2x
$y'$y $=$= $\ln2\ 2^x$ln2 2x

 

Practice question

Question 4

Consider the function $y=7^x$y=7x.

  1. Using the fact that $e^{\ln a}=a$elna=a, rewrite the function in terms of natural base $e$e.

  2. Hence determine $y'$y. Express the derivative in terms of the base $7$7.

    You may use the substitution $u=\left(\ln7\right)x$u=(ln7)x.

  3. Hence determine the exact gradient at $x=1$x=1.

Differentiating $y=a^{f\left(x\right)}$y=af(x)

Repeating the processes shown above we can find derivatives of functions in the form $y=a^{f\left(x\right)}$y=af(x).

Again we will use logarithm laws to determine the derivative of $y=a^{f\left(x\right)}$y=af(x) for the case when $a\ne e$ae.

$y$y $=$= $a^{f\left(x\right)}$af(x)

 

Taking the natural log of both sides gives:

$\ln y$lny $=$= $\ln a^{f\left(x\right)}$lnaf(x)

 

Using our log laws gives:

$\ln y$lny $=$= $f\left(x\right)\ln a$f(x)lna

 

Reversing the process, we get:

$e^{\ln y}$elny $=$= $e^{f\left(x\right)\ln a}$ef(x)lna

 

Which can be simplified to:

$y$y $=$= $e^{f\left(x\right)\ln a}$ef(x)lna
 

Taking the derivative of both sides using $\frac{d}{dx}e^{f\left(x\right)}=f'\left(x\right)e^{f\left(x\right)}$ddxef(x)=f(x)ef(x):

$\frac{dy}{dx}$dydx $=$= $\ln a\ f'(x)\ e^{f\left(x\right)\ln a}$lna f(x) ef(x)lna
 

But, remember $y=e^{f\left(x\right)\ln a}=a^{f\left(x\right)}$y=ef(x)lna=af(x), therefore:

$\frac{dy}{dx}$dydx $=$= $\ln a\ f'\left(x\right)\ a^{f\left(x\right)}$lna f(x) af(x)
 
Differentiating $y=a^{f\left(x\right)}$y=af(x)
$\frac{d}{dx}a^{f\left(x\right)}$ddxaf(x) $=$= $\ln a\ f'\left(x\right)\ a^{f\left(x\right)}$lna f(x) af(x)

Practice question

Question 5

Differentiate $y=5^{2-4x}+3$y=524x+3.

 

Outcomes

0606C14.3C

Use the derivatives of the standard functions e^x, ln x, together with constant multiples, sums and composite functions.

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