# C4 – Exam Tip #1

### Differentiating $y=a^x$

As part of the C4 content, you must know how to differentiate $y=a^x$ where $a$ is a constant.

The trick to finding $\frac{dy}{dx}$ when $y=a^x$ is first taking the natural logarithm of both sides of this equation:

$y=a^x$
$\Longrightarrow\hspace{7pt}\ln(y)=\ln\left(a^x\right)$

Then use the laws of logarithms to bring the $x$ outside:

$\ln(y)=\ln\left(a^x\right)$
$\Longrightarrow\hspace{7pt}\ln(y)=x\ln\left(a\right)$

We can then differentiate both sides of this equation (this means differentiating the left hand side implicitly):

$\ln(y)=x\ln(a)$
$\Longrightarrow\hspace{7pt}\frac{1}{y}\frac{dy}{dx}=\ln\left(a\right)$

Note that $ln(a)$ is a constant and so $x\ln(a)$

Finally, multiply both sides of the equation by $y$

$\frac{1}{y}\frac{dy}{dx}=\ln\left(a\right)$
$\Longrightarrow\hspace{7pt}\frac{dy}{dx}=y\ln\left(a\right)$

…and remember that $y=a^x$

$\frac{dy}{dx}=a^x\ln\left(a\right)$

0