NZ Level 8 (NZC) Level 3 (NCEA) [In development]
Basic derivative of e^x
Lesson

## Extending the Derivative

We have established the result that if $y=e^x$y=ex, then the derivative $\frac{\mathrm{d}y}{\mathrm{d}x}=e^x$dydx=ex.

We now use the chain rule to find the derivative of the more general function given by $y=e^{ax+b}$y=eax+b.

Setting $u=ax+b$u=ax+b so that $y=e^u$y=eu, we note that $\frac{\mathrm{d}u}{\mathrm{d}x}=a$dudx=a and $\frac{\mathrm{d}y}{\mathrm{d}u}=e^u$dydu=eu.

Thus we have: $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\times\frac{\mathrm{d}u}{\mathrm{d}x}=e^u\times a=ae^{ax+b}$dydx=dydu×dudx=eu×a=aeax+b

## Examples using the general rule

Here are a few examples using the generalised rule.  Why not try them all for yourself and see if you can get my derivative answers?

$f(x)$f(x) $f'(x)$f(x)
$e^{4x-7}$e4x7 $4e^{4x-7}$4e4x7
$-2e^{1-x}$2e1x $2e^{1-x}$2e1x
$x^2+\frac{4}{e^x}$x2+4ex $2x-\frac{4}{e^x}$2x4ex
$e^{\frac{x+3}{2}}$ex+32 $\frac{1}{2}e^{\frac{x+3}{2}}$12ex+32
$10x-\frac{e^{1-2x}}{2}$10xe12x2 $10+e^{1-2x}$10+e12x

#### A simple application

To find the equation of the tangent to the curve given by $y=e^{2x-5}$y=e2x5 at the point where $x=3$x=3, we first note that at $x=3$x=3$y=e^{2\left(3\right)-5}=e$y=e2(3)5=e

The derivative $\frac{\mathrm{d}y}{\mathrm{d}x}=2e^{2x-5}$dydx=2e2x5, and so at $x=3$x=3 , $\frac{\mathrm{d}y}{\mathrm{d}x}=2e$dydx=2e.

Using the point-gradient form, the equation of the tangent becomes:

 $y-y_1$y−y1​ $=$= $m\left(x-x_1\right)$m(x−x1​) $y-e$y−e $=$= $2e\left(x-3\right)$2e(x−3) $y-e$y−e $=$= $2ex-6e$2ex−6e $y$y $=$= $2ex-5e$2ex−5e

#### Worked Examples

##### QUESTION 1

Find the derivative of $y=e^{2x}+e^9+e^{-5x}$y=e2x+e9+e5x.

##### QUESTION 2

$f\left(x\right)=e^x$f(x)=ex and its tangent line at $x=0$x=0 are graphed on the coordinate axes.

1. Determine the gradient to the curve at $x=0$x=0.

2. Evaluate $f\left(0\right)$f(0).

3. Which of the following is true?

$f\left(0\right)\ne f'\left(0\right)$f(0)f(0)

A

$f\left(0\right)=f'\left(0\right)$f(0)=f(0)

B

$f\left(0\right)\ne f'\left(0\right)$f(0)f(0)

A

$f\left(0\right)=f'\left(0\right)$f(0)=f(0)

B

##### QUESTION 3

Consider the function $y=e^{ax}$y=eax, where $a$a is a constant.

1. Let $u=ax$u=ax.

Rewrite the function $y$y in terms of $u$u.

2. Determine $\frac{du}{dx}$dudx.

3. Hence determine $\frac{dy}{dx}$dydx.

4. Consider the function $f\left(x\right)=e^{2x}$f(x)=e2x.

State $f'\left(x\right)$f(x).

5. Consider the function $f\left(x\right)=e^{2x}$f(x)=e2x.

State $f'$f$\left(-3\right)$(3). Give your answer in exact form.

### 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