Question

question
if $y(x)=-9e^{x^{-9}}$, find $y(x)$.
select the correct answer below:
$y(x)=81x^{-9}e^{x^{-9}}$
$y(x)=81e^{x^{-10}}$
$y(x)=81x^{-10}e^{x^{-9}}$
$y(x)=-9e^{x^{-9}}$

Explanation:

Step1: Apply chain - rule

Let $u = x^{-9}$, then $y=-9e^{u}$. The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=-9e^{u}$, and the derivative of $u$ with respect to $x$ is $\frac{du}{dx}=- 9x^{-10}$.

Step2: Calculate $y'(x)$

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=(-9e^{u})\cdot(-9x^{-10})$. Substitute $u = x^{-9}$ back in, we get $y'(x)=81x^{-10}e^{x^{-9}}$.

Answer:

$y'(x)=81x^{-10}e^{x^{-9}}$