find h(x) where f(x) is an unspecified differentiable function.\nh(x)=7x^{4}f(x)\nchoose the correct answer…

find h(x) where f(x) is an unspecified differentiable function.\nh(x)=7x^{4}f(x)\nchoose the correct answer below.\na. h(x)=28x^{3}f(x)f(x)\nb. h(x)=7x^{4}f(x)+28x^{3}f(x)\nc. h(x)=28x^{3}f(x)\nd. h(x)=x^{2}f(x)(1 + 28x^{3})

find h(x) where f(x) is an unspecified differentiable function.\nh(x)=7x^{4}f(x)\nchoose the correct answer below.\na. h(x)=28x^{3}f(x)f(x)\nb. h(x)=7x^{4}f(x)+28x^{3}f(x)\nc. h(x)=28x^{3}f(x)\nd. h(x)=x^{2}f(x)(1 + 28x^{3})

Answer

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))