# Thread: Need help differentiating f(x) = a^x

1. ## Need help differentiating f(x) = a^x

How do I differentiate f(x) = a^x

The answer xa^(x-1) is not sufficient.

I think I'm supposed to substitute with e somehow to figure it out, but I'm not entirely sure how e works.

Can someone help me out? Please explain with detail.

2. ## Re: Need help differentiating f(x) = a^x

Originally Posted by TWN
How do I differentiate f(x) = a^x

The answer xa^(x-1) is not sufficient. CORRECT
Note that $\displaystyle a>0$ must be the case.

The answer is $\displaystyle f'(x)=a^x\ln(a)$. It is done with logarithmic differentiation.

3. ## Re: Need help differentiating f(x) = a^x

Can you explain in detail the process you underwent to reach that answer please?

4. ## Re: Need help differentiating f(x) = a^x

$\displaystyle y = a^x$

$\displaystyle \ln{y} = \ln{a^x}$

$\displaystyle \ln{y} = x \ln{a}$

$\displaystyle \frac{d}{dx} \left[\ln{y} = x \ln{a} \right]$

$\displaystyle \frac{y'}{y} = \ln{a}$

$\displaystyle y' = y \cdot \ln{a}$

$\displaystyle y' = a^x \cdot \ln{a}$