Question

Find the derivative of the function. $$ f(x)=\frac{2}{\sqrt[3]{x}}+5 \cos x $$

Short answer

The derivative of the function \(f(x)=\frac{2}{\sqrt[3]{x}}+5 \cos x\) is \(f'(x) = -2/3x^{-4/3} - 5\sin(x)\).

Step-by-step solution

Rewrite the function

To prepare for differentiation, rewrite the function in a form that will make the differentiation process simpler. Rewriting the function using the power rule leads to \(f(x) = 2x^{-1/3} + 5\cos(x)\). This form allows easier application of our rules of differentiation.

Differentiate the first term

Apply the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\). Then, the derivative of \(2x^{-1/3}\) is \(-2/3 x^{-1/3-1} = -2/3x^{-4/3}\).

Differentiate the second term

The derivative of \(\cos(x)\) is \(-\sin(x)\). Hence, the derivative of \(5\cos(x)\) is \(5(-\sin(x)) = -5\sin(x)\).

Combine the results

Combine the results obtained from step 2 and step 3 to write the derivative of the total function as \(f'(x) = -2/3x^{-4/3} - 5\sin(x)\).