Textbook Question
Evaluate the derivative of the following functions.
f(t) = ln (sin-1 t2)
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 29
Find and simplify the derivative of the following functions.
y = (3t−1)(2t−2)-1
Verified step by step guidance1
Step 1: Identify the function y = (3t - 1)(2t - 2)^{-1} as a product of two functions, u(t) = 3t - 1 and v(t) = (2t - 2)^{-1}.
Step 2: Use the product rule for differentiation, which states that if y = u(t)v(t), then y' = u'(t)v(t) + u(t)v'(t).
Step 3: Differentiate u(t) = 3t - 1 to find u'(t). The derivative of 3t is 3, and the derivative of -1 is 0, so u'(t) = 3.
Step 4: Differentiate v(t) = (2t - 2)^{-1} using the chain rule. Rewrite v(t) as (2t - 2)^{-1} = (2t - 2)^{-1} = (2t - 2)^{-1}. The derivative of (2t - 2)^{-1} is -1(2t - 2)^{-2} times the derivative of (2t - 2), which is 2.
Step 5: Substitute u(t), u'(t), v(t), and v'(t) into the product rule formula to find y'. Simplify the expression to obtain the derivative of y.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.
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Derivatives
Product Rule
The product rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions u(t) and v(t), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are multiplied together, as in the given function y = (3t−1)(2t−2)⁻¹.
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05:18The Product Rule
Quotient Rule
The quotient rule is used to differentiate functions that are expressed as the ratio of two other functions. If y = u(t)/v(t), the derivative is given by (u'v - uv')/v². This rule is particularly relevant for the function in the question, as it involves a term raised to a negative exponent, which can be interpreted as a quotient.
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Related Practice
Textbook Question
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x)= 1/(2x + 1); P (0,1)
Textbook Question
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = √(x - 1); P (2,1)
Textbook Question
Consider the following cost functions.
b. Determine the average cost and the marginal cost when x=a.
C(x) = 500+0.02x, 0≤x≤2000, a=1000
Textbook Question
Consider the following cost functions.
c. Interpret the values obtained in part (b).
C(x) = 1000+0.1x, 0≤x≤5000, a=2000
Textbook Question
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = x3; P (1,1)
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