Question

Say that f and h are functions and that f(3) = 2, f'(3) = -2, h(3) = 1 and h'(3) = 4. Find p'(3) where p(x) = f(x) / (x²)h(x).

Answer 1

To find p'(3) for the function p(x) = f(x) / (x²)h(x), we use the quotient rule of differentiation with the given values of f(3), f'(3), h(3), and h'(3). Simplifying the derived expression leads to p'(3) = -22 / 27.

Step-by-step explanation:

To find p'(3) where p(x) = f(x) / (x²)h(x), we'll use the product and quotient rules for differentiation. Given f(3) = 2, f'(3) = -2, h(3) = 1, and h'(3) = 4, we'll differentiate p(x) which is a quotient of two functions.

First, let's set g(x) = x², which implies g'(x) = 2x. Now, p(x) = f(x) / (g(x)h(x)), and we can apply the quotient rule:

p'(x) = [(g(x)h(x))f'(x) - f(x)(g'(x)h(x) + g(x)h'(x))] / (g(x)h(x))²

Substituting the given values:

p'(3) = [((3)²·(1))(-2) - (2)((2·(3))(1) + (3)²(4))] / [((3)²·(1))²]

Then we simplify:

p'(3) = [(9·(-2)) - (2·(6) + 9·(4))] / 81

p'(3) = (-18 - (12 + 36)) / 81

p'(3) = (-18 - 48) / 81

p'(3) = -66 / 81

p'(3) = -22 / 27