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)

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