Question

H(x)=f(x)g(x). if f(x)=lnx and g(x)=4x^2−2x 2, what is h′(x)?

Answer 1

To find h′(x), the derivative of the function h(x) = f(x)g(x), we need to use the product rule. Using the given functions f(x) = ln(x) and g(x) = 4x^2−2x^2, we can find the derivatives and substitute them into the product rule equation.

Step-by-step explanation:

To find h′(x), the derivative of the function h(x)= f(x)g(x), we need to use the product rule. The product rule states that if we have two functions, f(x) and g(x), then the derivative of their product is given by:

h′(x) = f(x)g′(x) + f′(x)g(x)

Using the given functions, f(x) = ln(x) and g(x) = 4x^2−2x^2, we can find the derivatives:

f′(x) = 1/x (since the derivative of ln(x) with respect to x is 1/x)

g′(x) = 8x−4 (by applying the power rule and constant multiple rule)

Substituting these derivatives into the product rule equation, we get:

h′(x) = ln(x) * (8x−4) + (1/x) * (4x^2−2x)

Simplifying further gives:

h′(x) = (8xln(x)−4ln(x)) + (4x−2)