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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.57b
Chapter 3, Problem 3.57b

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.


x ƒ(x) g(x) ƒ'(x) g'(x)
0 1 1 -3 1/2
1 3 5 1/2 -4


Find the first derivatives of the following combinations at the given value of x.


b. ƒ(x)g²(x), x = 0

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To find the derivative of the function ƒ(x)g²(x) at x = 0, we need to apply the product rule and the chain rule. The product rule states that the derivative of a product of two functions u(x) and v(x) is given by u'(x)v(x) + u(x)v'(x).
In this case, let u(x) = ƒ(x) and v(x) = g²(x). We need to find the derivative of v(x) using the chain rule. The chain rule states that the derivative of a composite function h(g(x)) is h'(g(x))g'(x).
Since v(x) = g²(x), we can express it as h(g(x)) where h(x) = x². The derivative h'(x) is 2x, so the derivative of v(x) = g²(x) is 2g(x)g'(x).
Now, apply the product rule: the derivative of ƒ(x)g²(x) is ƒ'(x)g²(x) + ƒ(x)(2g(x)g'(x)).
Substitute the given values at x = 0: ƒ'(0) = -3, g(0) = 1, g'(0) = 1/2, and ƒ(0) = 1. Calculate ƒ'(0)g²(0) + ƒ(0)(2g(0)g'(0)) using these values.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Product Rule

The Product Rule is a fundamental principle in calculus used to differentiate products of two functions. It states that if you have two functions, ƒ(x) and g(x), the derivative of their product is given by ƒ'(x)g(x) + ƒ(x)g'(x). This rule is essential for finding the derivative of combinations of functions, such as ƒ(x)g²(x), as it allows us to apply the differentiation process systematically.
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The Product Rule

Chain Rule

The Chain Rule is another critical differentiation technique used when dealing with composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x is the derivative of y with respect to u multiplied by the derivative of u with respect to x. In the context of g²(x), the Chain Rule helps us differentiate the square of the function g(x) effectively.
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Intro to the Chain Rule

Evaluating Derivatives at Specific Points

Evaluating derivatives at specific points involves substituting the given x-value into the derivative expression to find the slope of the function at that point. In this problem, we need to compute the derivatives of ƒ(x)g²(x) at x = 0, which requires using the values of ƒ(0), g(0), ƒ'(0), and g'(0) provided in the table. This step is crucial for obtaining the final numerical result.
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Critical Points
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