Question

find h(x) where f(x) is an unspecified differentiable function.
h(x)=7x^{4}f(x)
choose the correct answer below.
a. h(x)=28x^{3}f(x)f(x)
b. h(x)=7x^{4}f(x)+28x^{3}f(x)
c. h(x)=28x^{3}f(x)
d. h(x)=x^{2}f(x)(1 + 28x^{3})

Explanation:

Step1: Apply product - rule

The product - rule states that if \(h(x)=u(x)v(x)\), then \(h^{\prime}(x)=u^{\prime}(x)v(x)+u(x)v^{\prime}(x)\). Here, \(u(x) = 7x^{4}\) and \(v(x)=f(x)\).

Step2: Differentiate \(u(x)\)

Differentiate \(u(x)=7x^{4}\) with respect to \(x\). Using the power - rule \((x^{n})^\prime=nx^{n - 1}\), we have \(u^{\prime}(x)=\frac{d}{dx}(7x^{4})=7\times4x^{3}=28x^{3}\), and \(v^{\prime}(x)=f^{\prime}(x)\).

Step3: Calculate \(h^{\prime}(x)\)

Substitute \(u(x)\), \(u^{\prime}(x)\), \(v(x)\), and \(v^{\prime}(x)\) into the product - rule formula. \(h^{\prime}(x)=u^{\prime}(x)v(x)+u(x)v^{\prime}(x)=28x^{3}f(x)+7x^{4}f^{\prime}(x)\).

Answer:

B. \(h^{\prime}(x)=7x^{4}f^{\prime}(x)+28x^{3}f(x)\)