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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 17
Capitolo 3, Problema 17

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = sin⁵x

Guida verificata passo dopo passo
1
Step 1: Identify the composite function structure. The given function is y = \(\sin\)^5(x), which can be rewritten as y = (\(\sin\)(x))^5.
Step 2: Determine the inner function u = g(x). In this case, the inner function is u = \(\sin\)(x).
Step 3: Determine the outer function y = f(u). The outer function is y = u^5.
Step 4: Differentiate the outer function with respect to u. The derivative of y = u^5 with respect to u is \(\frac{dy}{du}\) = 5u^4.
Step 5: Differentiate the inner function with respect to x. The derivative of u = \(\sin\)(x) with respect to x is \(\frac{du}{dx}\) = \(\cos\)(x).

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Composite Functions

A composite function is formed when one function is applied to the result of another function. In the context of the question, we need to identify an inner function g(x) and an outer function f(u) such that the overall function can be expressed as y = f(g(x)). Understanding how to break down a function into its components is essential for differentiation.
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Evaluate Composite Functions - Special Cases

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if y = f(g(x)), then the derivative dy/dx can be calculated as dy/dx = f'(g(x)) * g'(x). This rule allows us to find the derivative of complex functions by differentiating 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 y = x^n, then dy/dx = n*x^(n-1). In the given function y = sin⁵(x), recognizing that this is a power function of sin(x) is crucial. Applying the power rule in conjunction with the chain rule will help in finding the derivative of the composite function effectively.
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Power Rules
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Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

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5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = (5x²+11x)^4/3

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On what intervals is the speed increasing?

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Determine the acceleration of the object when its velocity is zero.

f(t) = t2 - 4t; 0 ≤ t ≤ 5

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