Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 3 - Derivatives

Problema 3.6.41

Capitolo 3, Problema 3.6.41

In Exercises 41–58, find dy/dt.

y = sin²(πt − 2)

Guida verificata passo dopo passo

Identify the function y = sin²(πt − 2). This can be rewritten as y = (sin(πt − 2))² to make it easier to differentiate.

Recognize that you need to use the chain rule to differentiate y with respect to t. The chain rule states that if you have a composite function, the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Differentiate the outer function u² with respect to u, where u = sin(πt − 2). The derivative of u² with respect to u is 2u.

Differentiate the inner function sin(πt − 2) with respect to t. The derivative of sin(πt − 2) is cos(πt − 2) multiplied by the derivative of the inside function πt − 2, which is π.

Combine the results from the previous steps using the chain rule: dy/dt = 2(sin(πt − 2)) * cos(πt − 2) * π.

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Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(t)) is composed of two functions, the derivative dy/dt is found by multiplying the derivative of the outer function f with respect to the inner function g by the derivative of the inner function g with respect to t. This is essential for differentiating y = sin²(πt − 2).

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Intro to the Chain Rule

Derivative of Sine Function

Understanding the derivative of the sine function is crucial, as it forms the basis for differentiating trigonometric expressions. The derivative of sin(x) with respect to x is cos(x). In the context of y = sin²(πt − 2), recognizing that the derivative of sin(πt − 2) is cos(πt − 2) is key to applying the chain rule effectively.

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Derivatives of Sine & Cosine

Power Rule

The power rule is a basic differentiation rule used when differentiating functions of the form u^n, where u is a function of t. It states that the derivative is n*u^(n-1) * du/dt. For y = sin²(πt − 2), this rule helps in differentiating the squared term, where n is 2, and u is sin(πt − 2), requiring the use of the chain rule for du/dt.

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Power Rules

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Heating a plate When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm/min. At what rate is the plate’s area increasing when the radius is 50 cm?

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Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

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Using the Alternative Formula for Derivatives

Use the formula

f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)