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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 3.6.75g
Capitolo 3, Problema 3.6.75g

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.


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Find the derivatives with respect to x of the following combinations at the given value of x.


g. 1 / g²(x), x = 3

Guida verificata passo dopo passo
1
Step 1: Recall the derivative rule for the reciprocal of a function. If h(x) = 1 / g²(x), then h'(x) can be found using the chain rule and the power rule. Specifically, h'(x) = -2 * g(x)^(-3) * g'(x).
Step 2: Substitute the given value of x = 3 into the formula. From the table, g(x) = -4 and g'(x) = 5 at x = 3.
Step 3: Replace g(x) and g'(x) in the derivative formula. This gives h'(x) = -2 * (-4)^(-3) * 5.
Step 4: Simplify the expression step by step. First, calculate (-4)^(-3), which is the reciprocal of (-4)^3. Then multiply by -2 and g'(x).
Step 5: The simplified expression will give the derivative h'(x) at x = 3. Ensure all calculations are consistent with the rules of exponents and multiplication.

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Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this problem, the chain rule helps differentiate expressions involving powers of functions, such as g²(x), by considering the derivative of the outer function and multiplying it by the derivative of the inner function.
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Intro to the Chain Rule

Power Rule

The power rule is a basic differentiation rule that states if f(x) = x^n, then f'(x) = n*x^(n-1). This rule is essential for finding the derivative of functions raised to a power, such as g²(x). In the context of the problem, it helps simplify the differentiation process by providing a straightforward method to handle powers of functions.
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Power Rules

Quotient Rule

The quotient rule is used to differentiate functions that are ratios of two other functions. It states that if h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) - f(x)g'(x))/g²(x). In this problem, the quotient rule is applied to find the derivative of 1/g²(x), which involves differentiating a function in the form of a reciprocal, requiring careful application of the rule to ensure accuracy.
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The Quotient Rule
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Find the derivatives with respect to x of the following combinations at the given value of x.


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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)

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


g. ƒ(x + g(x)), x = 0

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