Question

find the derivative of $f(x) = -4x^5 - 11\sin x + \frac{7}{x^5}$.$\bigcirc$ $f(x) = -20x^4 + 11\cos x + \frac{7}{5x^4}LXB0\bigcirc$ $f(x) = -20x^4 - 11\cos x + \frac{7}{5x^4}$$\bigcirc$ $f(x) = -20x^4 - 11\cos x - \frac{35}{x^6}$

Explanation:

Step1: Rewrite the rational term

Rewrite $\frac{7}{x^5}$ as $7x^{-5}$, so $f(x) = -4x^5 -11\sin x + 7x^{-5}$.

Step2: Differentiate term by term

Apply power rule $\frac{d}{dx}(x^n)=nx^{n-1}$ and derivative of sine: $\frac{d}{dx}(\sin x)=\cos x$.

• Derivative of $-4x^5$: $-4\times5x^{5-1}=-20x^4$
• Derivative of $-11\sin x$: $-11\cos x$
• Derivative of $7x^{-5}$: $7\times(-5)x^{-5-1}=-35x^{-6}=-\frac{35}{x^6}$

Step3: Combine the results

Combine the derivatives of all terms.

Answer:

$f'(x)= -20x^{4}-11\cos x-\frac{35}{x^{6}}$ (the fourth option)