Question

find the derivative of the following function. y = 3 cos x+2e^x. dy/dx =

Explanation:

Step1: Apply sum - rule of derivatives

$\frac{d}{dx}(u + v)=\frac{du}{dx}+\frac{dv}{dx}$, where $u = 3\cos x$ and $v = 2e^{x}$. So $\frac{dy}{dx}=\frac{d}{dx}(3\cos x)+\frac{d}{dx}(2e^{x})$.

Step2: Use constant - multiple rule

The constant - multiple rule is $\frac{d}{dx}(cf(x))=c\frac{d}{dx}(f(x))$. So $\frac{d}{dx}(3\cos x)=3\frac{d}{dx}(\cos x)$ and $\frac{d}{dx}(2e^{x})=2\frac{d}{dx}(e^{x})$.

Step3: Recall derivative formulas

We know that $\frac{d}{dx}(\cos x)=-\sin x$ and $\frac{d}{dx}(e^{x})=e^{x}$. Then $3\frac{d}{dx}(\cos x)=3(-\sin x)=- 3\sin x$ and $2\frac{d}{dx}(e^{x})=2e^{x}$.

Step4: Combine results

$\frac{dy}{dx}=-3\sin x + 2e^{x}$.

Answer:

$-3\sin x + 2e^{x}$