Question

compute the derivative of the following function. f(x)=5xe^{6x} f(x)=□

Explanation:

Step1: Apply product - rule

The product - rule states that if \(y = u\cdot v\), then \(y^\prime=u^\prime v + uv^\prime\). Here, let \(u = 5x\) and \(v = e^{6x}\).

Step2: Find the derivative of \(u\)

The derivative of \(u = 5x\) with respect to \(x\) is \(u^\prime=\frac{d}{dx}(5x)=5\).

Step3: Find the derivative of \(v\)

Using the chain - rule, if \(v = e^{6x}\), let \(t = 6x\), then \(\frac{dv}{dx}=\frac{dv}{dt}\cdot\frac{dt}{dx}\). Since \(\frac{d}{dt}(e^{t}) = e^{t}\) and \(\frac{dt}{dx}=6\), we have \(v^\prime=\frac{d}{dx}(e^{6x})=6e^{6x}\).

Step4: Calculate \(f^\prime(x)\)

By the product - rule \(f^\prime(x)=u^\prime v+uv^\prime\). Substitute \(u = 5x\), \(u^\prime = 5\), \(v = e^{6x}\), and \(v^\prime = 6e^{6x}\) into the formula: \(f^\prime(x)=5\cdot e^{6x}+5x\cdot6e^{6x}=5e^{6x}(1 + 6x)\).

Answer:

\(5e^{6x}(1 + 6x)\)