Question

for the function ( y=3x^e+2e^x ), find ( \frac{dy}{dx} ) by rules. answer: ( \frac{dy}{dx}= ) .

Explanation:

Step1: Differentiate \(3x^e\)

Use the power rule \(\frac{d}{dx}(x^n)=nx^{n - 1}\). Here \(n = e\), so \(\frac{d}{dx}(3x^e)=3e x^{e - 1}\).

Step2: Differentiate \(2e^x\)

Use the exponential rule \(\frac{d}{dx}(e^x)=e^x\), so \(\frac{d}{dx}(2e^x)=2e^x\).

Step3: Sum the derivatives

Add the results from Step1 and Step2: \(\frac{dy}{dx}=3e x^{e - 1}+2e^x\).

Answer:

\(3e x^{e - 1}+2e^x\)