Question

calculate the derivative of the following function. y=(3 - e^x)^4 dy/dx = □

Explanation:

Step1: Apply chain - rule

Let $u = 3 - e^{x}$, 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 $u = 3 - e^{x}$ with respect to $x$. Since $\frac{d}{dx}(3)=0$ and $\frac{d}{dx}(e^{x})=e^{x}$, we have $\frac{du}{dx}=-e^{x}$.

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

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

Answer:

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