Question

compute the derivative of the following function.
f(x)=6x + 7xe^{x}
f(x)=\square

Explanation:

Step1: Apply sum rule

The sum rule states that if \(f(x)=u(x)+v(x)\), then \(f^{\prime}(x)=u^{\prime}(x)+v^{\prime}(x)\). Let \(u(x) = 6x\) and \(v(x)=7xe^{x}\). The derivative of \(u(x)\) is \(u^{\prime}(x)=\frac{d}{dx}(6x)=6\).

Step2: Apply product rule for \(v(x)\)

The product rule states that if \(y = ab\), where \(a = 7x\) and \(b=e^{x}\), then \(y^{\prime}=a^{\prime}b + ab^{\prime}\).

• \(a^{\prime}=\frac{d}{dx}(7x)=7\)
• \(b^{\prime}=\frac{d}{dx}(e^{x})=e^{x}\)

So \(v^{\prime}(x)=7e^{x}+7xe^{x}\)

Step3: Combine results

\(f^{\prime}(x)=u^{\prime}(x)+v^{\prime}(x)=6 + 7e^{x}+7xe^{x}=6+7e^{x}(1 + x)\)

Answer:

\(6 + 7e^{x}(x + 1)\)