Domanda del libro di testo
Evaluate and simplify y'.
y = 2x√2
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.R.29
9–61. Evaluate and simplify y'.
y = tan^−1 √t²−1
Guida verificata passo dopo passo1
First, identify the function y in terms of t. Here, y = tan^(-1)(√(t²−1)).
To find y', the derivative of y with respect to t, use the chain rule. The chain rule states that if a function y = f(g(t)), then y' = f'(g(t)) * g'(t).
Recognize that y = tan^(-1)(u) where u = √(t²−1). The derivative of tan^(-1)(u) with respect to u is 1/(1+u²).
Next, find the derivative of u = √(t²−1) with respect to t. Use the chain rule again: if u = (t²−1)^(1/2), then u' = (1/2)(t²−1)^(-1/2) * 2t = t/√(t²−1).
Combine the derivatives using the chain rule: y' = (1/(1+u²)) * (t/√(t²−1)). Substitute u = √(t²−1) back into the expression to simplify y'.
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Derivative of Inverse Functions
The derivative of an inverse function, such as the arctangent function, can be found using the formula: if y = tan^(-1)(u), then dy/dx = 1/(1 + u^2) * du/dx. This concept is essential for differentiating functions that involve inverse trigonometric functions.
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Derivatives of Inverse Sine & Inverse Cosine
Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. If a function y = f(g(t)), the derivative is given by dy/dt = f'(g(t)) * g'(t). This rule is crucial for evaluating the derivative of y = tan^(-1)(√(t² - 1)) since it involves a composition of functions.
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05:02Intro to the Chain Rule
Simplifying Derivatives
After finding the derivative, simplifying the expression is often necessary to make it more manageable or interpretable. This may involve factoring, combining like terms, or rationalizing the denominator. Understanding how to simplify derivatives is key to presenting the final answer clearly.
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6:36Simplifying Trig Expressions
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