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Chapter 3: Problem 1

How do you find the derivative of the product of two functions that are differentiable at a point?

Short Answer

Expert verified
Question: Find the derivative of the product of two differentiable functions, f(x) and g(x), at point x_0. Answer: To find the derivative of the product of two differentiable functions, f(x) and g(x) at point x_0, apply the Product Rule and substitute the derivatives of the individual functions and the given point into the formula: h'(x_0) = (f'(x_0))g(x_0) + f(x_0)(g'(x_0))

Step by step solution

01

Identify the given functions

Given two differentiable functions, let these functions be represented as f(x) and g(x).
02

Differentiate each function separately

Find the derivatives of the two functions, f'(x) and g'(x). f'(x) = derivative of f(x) g'(x) = derivative of g(x)
03

Apply the Product Rule for differentiation

According to the Product Rule, if we have a composite function h(x) = f(x)g(x), the derivative of h(x) with respect to x, h'(x) can be found by: h'(x) = f'(x)g(x) + f(x)g'(x)
04

Substitute the derivatives, f'(x) and g'(x) into the formula

Now, substitute the derivatives of the individual functions, f'(x) and g'(x) into the Product Rule formula: h'(x) = (f'(x))g(x) + f(x)(g'(x))
05

Evaluate the differentiation at the given point

Substitute the value of the given point x, into the expression h'(x) determined in step 4 to get the derivative of the product at that specific point: h'(x_0) = (f'(x_0))g(x_0) + f(x_0)(g'(x_0)) This will yield the derivative of the product of the two functions at the given point x_0.

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Most popular questions from this chapter

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