if $f(x)=9^{-2x}$ what is $f(x)=$

Explanation:

Step1: Recall exponential - derivative formula

The derivative of $a^{u}$ with respect to $x$ is $a^{u}\ln(a)\cdot u'$, where $a > 0,a\neq1$ and $u$ is a function of $x$. Here $a = 9$ and $u=-2x$.

Step2: Find the derivative of $u$

If $u = - 2x$, then $u'=-2$.

Step3: Calculate the first - derivative

Using the formula $\frac{d}{dx}(a^{u})=a^{u}\ln(a)\cdot u'$, we have $f'(x)=9^{-2x}\ln(9)\cdot(-2)=-2\ln(9)\cdot9^{-2x}$.

Step4: Find the second - derivative

Now we find the derivative of $f'(x)$. Let $y = f'(x)=-2\ln(9)\cdot9^{-2x}$. Here $a = 9$ and $u=-2x$ again. The derivative of $y$ with respect to $x$ is $y'=-2\ln(9)\cdot9^{-2x}\ln(9)\cdot(-2)$.

Step5: Simplify the second - derivative

$f''(x)=4(\ln(9))^{2}\cdot9^{-2x}$.

Answer:

$4(\ln(9))^{2}\cdot9^{-2x}$