Need help differentiating f(x) = a^x

**TWN** · May 15th 2012, 02:20 PM

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.

**Plato** · May 15th 2012, 02:38 PM

Originally Posted by TWN

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

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

Note that $a>0$ must be the case.

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

**TWN** · May 15th 2012, 02:48 PM

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

**skeeter** · May 15th 2012, 03:26 PM

$y = a^x$

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

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

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

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

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

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

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