Question

differentiate the following function.

$f(x)=e^{5(x^{2}+4)^{2}}$

$f(x)=\square$

Explanation:

Step1: Apply the chain rule

The chain rule states that if \(y = e^{u}\), then \(y^\prime=e^{u}\cdot u^\prime\). Let \(u = 5(x^{2}+4)^{2}\). First, find the derivative of \(u\) with respect to \(x\).

Step2: Differentiate \(u = 5(x^{2}+4)^{2}\)

Use the chain rule again. Let \(v=x^{2}+4\), so \(u = 5v^{2}\). The derivative of \(u\) with respect to \(v\) is \(u_v^\prime=10v\), and the derivative of \(v\) with respect to \(x\) is \(v_x^\prime = 2x\). Then \(u^\prime=10v\cdot2x=10(x^{2}+4)\cdot2x = 20x(x^{2}+4)\).

Step3: Differentiate \(y = e^{u}\)

Since \(y = e^{u}\) and \(u = 5(x^{2}+4)^{2}\), by the chain rule \(y^\prime=e^{u}\cdot u^\prime\). Substitute \(u\) and \(u^\prime\) into the formula: \(y^\prime=e^{5(x^{2}+4)^{2}}\cdot20x(x^{2}+4)\).

Answer:

\(20x(x^{2}+4)e^{5(x^{2}+4)^{2}}\)