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