NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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Differentiating exponential functions
Lesson

A more general result

In previous work we have shown that if $y=e^{ax+b}$y=eax+b, then $\frac{dy}{dx}=ae^{ax+b}$dydx=aeax+b.

This result can be extended to functions of the form $y=e^{f\left(x\right)}$y=ef(x).

Since by letting $u=f\left(x\right)$u=f(x), we have $\frac{du}{dx}=f'\left(x\right)$dudx=f(x) and:

$\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$dydx=dydu×dudx

$\frac{dy}{dx}=e^u\times u'$dydx=eu×u

$\frac{dy}{dx}=e^{f\left(x\right)}\times f'\left(x\right)$dydx=ef(x)×f(x)

Thus if $y=e^{f\left(x\right)}$y=ef(x), then $\frac{dy}{dx}=f'\left(x\right)e^{f\left(x\right)}$dydx=f(x)ef(x).

Examples

The following seven examples show how to determine derivatives involving the exponential function. Note the use of the product quotient and chain rule through the examples:

Example 1 (Splitting fractions)
$y$y $=$= $\frac{e^{2x}-e^{-x}}{e^x}$e2xexex
  $=$= $\frac{e^{2x}}{e^x}-\frac{e^{-x}}{e^x}$e2xexexex
  $=$= $e^x-e^{-2x}$exe2x
$\therefore\frac{dy}{dx}$dydx $=$= $e^x+2e^{-2x}$ex+2e2x
     
Example 2 (for functions of the form $e^u$eu)
$y$y $=$= $e^{3x^2-x-2}$e3x2x2
$\therefore\frac{dy}{dx}$dydx $=$= $\left(6x-1\right)e^{3x^2-x-2}$(6x1)e3x2x2
     
Example 3 (Using the product rule $\left(uv\right)'=uv'+vu'$(uv)=uv+vu)
$y$y $=$= $e^{x^2}\left(1-x+x^3\right)$ex2(1x+x3)
$\therefore\frac{dy}{dx}$dydx $=$= $e^{x^2}\left(3x^2-1\right)+\left(1-x+x^3\right)2xe^{x^2}$ex2(3x21)+(1x+x3)2xex2
  $=$= $3x^2e^{x^2}-e^{x^2}+2xe^{x^2}-2x^2e^{x^2}+2x^4e^{x^2}$3x2ex2ex2+2xex22x2ex2+2x4ex2
  $=$= $e^{x^2}\left(3x^2-1+2x-2x^2+2x^4\right)$ex2(3x21+2x2x2+2x4)
  $=$= $e^{x^2}\left(2x^4+x^2+2x-1\right)$ex2(2x4+x2+2x1)
     
Example 4 (chain rule within a product rule)
$y$y $=$= $e^x\sqrt{1-2x}$ex12x
$y$y $=$= $e^x\left(1-2x\right)^{\frac{1}{2}}$ex(12x)12
$\therefore\frac{dy}{dx}$dydx $=$= $e^x\left[\frac{1}{2}\left(1-2x\right)^{-\frac{1}{2}}\times-2\right]+e^x\left(1-2x\right)^{\frac{1}{2}}$ex[12(12x)12×2]+ex(12x)12
  $=$= $e^x\left(\frac{-1}{\sqrt{1-2x}}\right)+e^x\sqrt{1-2x}$ex(112x)+ex12x
  $=$= $e^x\left(\sqrt{1-2x}-\frac{1}{\sqrt{1-2x}}\right)$ex(12x112x)
     
Example 5 (quotient rule)
$y$y $=$= $\frac{e^{x^2}}{e^x+5}$ex2ex+5
$\therefore\frac{dy}{dx}$dydx $=$= $\frac{\left(e^x+5\right)\left(2xe^x\right)-e^{x^2}\left(e^x\right)}{\left(e^x+5\right)^2}$(ex+5)(2xex)ex2(ex)(ex+5)2
  $=$= $\frac{e^x\left(2xe^x+10x-e^{x^2}\right)}{\left(e^x+5\right)^2}$ex(2xex+10xex2)(ex+5)2
     
Example 6 (The chain rule)
$y$y $=$= $\left(e^x+x\right)^{\frac{3}{4}}$(ex+x)34
$\therefore\frac{dy}{dx}$dydx $=$= $\frac{3}{4}\left(e^x+x\right)^{\frac{3}{4}-1}\times\left(e^x+1\right)$34(ex+x)341×(ex+1)
  $=$= $\frac{3}{4}\left(e^x+1\right)\left(e^x+x\right)^{-\frac{1}{4}}$34(ex+1)(ex+x)14
  $=$= $\frac{3\left(e^x+1\right)}{4\sqrt[4]{e^x+x}}$3(ex+1)44ex+x
     
Example 7 ($a^x$ax)

To find the derivative of $y=a^x$y=ax where the base is not necessarily $e$e, we first note that the number $a$a can be written as $e^{\log_e\left(a\right)}$eloge(a). Think of it as reverse operations. Taking the logarithm of $a$a, base $e$e, and then using that answer as the power that $e$e is raised to, simply returns you back to $a$a.

Thus the function $y=a^x$y=ax can be written $e^{\log_e\left(a\right)\times x}$eloge(a)×x or simply $y=e^{x\log_e\left(a\right)}$y=exloge(a).

Thus, $\frac{dy}{dx}=\left(\log_ea\right)e^{x\log_ea}$dydx=(logea)exlogea and this simplifies to $\frac{dy}{dx}=\left(\log_ea\right)e^a$dydx=(logea)ea

The result shows for example, that if $y=2^x$y=2x, then $\frac{dy}{dx}=\log_e\left(2\right)\times2^x=2^x\left(\ln2\right)$dydx=loge(2)×2x=2x(ln2). Similarly for $y=3^x$y=3x$\frac{dy}{dx}=3^x\left(\ln3\right)$dydx=3x(ln3).  

More Worked Examples

QUESTION 1

Find the derivative of $y=e^{x^2+3x+5}$y=ex2+3x+5.

QUESTION 2

Find the derivative of $y=e^{-4x}\sqrt{x+2}$y=e4xx+2.

QUESTION 3

Find the derivative of $y=\left(e^{2x}+5x\right)^{\frac{3}{4}}$y=(e2x+5x)34.

Outcomes

M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

91578

Apply differentiation methods in solving problems

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