Question

calculate the derivative of the following function.
y = (9 - e^x)^4
dy/dx = □

Explanation:

Step1: Apply chain - rule

Let $u = 9 - e^{x}$, then $y = u^{4}$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.

Step2: Calculate $\frac{dy}{du}$

Differentiate $y = u^{4}$ with respect to $u$. Using the power - rule $\frac{d}{du}(u^{n})=nu^{n - 1}$, we get $\frac{dy}{du}=4u^{3}$.

Step3: Calculate $\frac{du}{dx}$

Differentiate $u = 9 - e^{x}$ with respect to $x$. The derivative of a constant is 0 and the derivative of $e^{x}$ is $e^{x}$, so $\frac{du}{dx}=-e^{x}$.

Step4: Substitute $u$ and find $\frac{dy}{dx}$

Substitute $u = 9 - e^{x}$ into $\frac{dy}{du}$ and multiply by $\frac{du}{dx}$. We have $\frac{dy}{dx}=4(9 - e^{x})^{3}\cdot(-e^{x})=-4e^{x}(9 - e^{x})^{3}$.

Answer:

$-4e^{x}(9 - e^{x})^{3}$