Question

Find the derivative of the function. $$ f(x)=x^{-2}-2 e^{x} $$

Short answer

The derivative of \(f(x) = x^{-2} - 2e^{x}\) is \(-2x^{-3} - 2e^{x}\).

Step-by-step solution

Differentiate \(x^{-2}\)

The rule for differentiating \(x^n\) where \(n\) is any real number is \(nx^{n-1}\). Therefore, the derivative of \(x^{-2}\) is \(-2x^{-3}\).

Differentiate \(-2e^{x}\)

The derivative of \(e^{x}\) is \(e^{x}\) itself, and any constant multiplied by the function is also multiplied by the derivative. Hence, the derivative of \(-2e^{x}\) is \(-2e^{x}\).

Write the final derivative

Combining the results from step 1 and step 2, the derivative of \(f(x)\) is \(-2x^{-3} - 2e^{x}\).