Question

differentiate.
$f(x) = x^5 - 9e^{2x}$
$f(x) = \square$

Explanation:

Step1: Differentiate \(x^5\)

Power rule: \(\frac{d}{dx}(x^n)=nx^{n-1}\), so \(\frac{d}{dx}(x^5)=5x^4\)

Step2: Differentiate \(-9e^{2x}\)

Chain rule: \(\frac{d}{dx}(e^{ax})=ae^{ax}\), so \(\frac{d}{dx}(-9e^{2x})=-9\times2e^{2x}=-18e^{2x}\)

Step3: Combine results

Add derivatives of both terms: \(5x^4 - 18e^{2x}\)

Answer:

\(5x^4 - 18e^{2x}\)