Textbook Question
Tangent lines Find an equation of the line tangent to the graph of f at the given point.
f(x) = sin−1(x/4); (2,π/6)
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 3.4.34
Derivatives Find and simplify the derivative of the following functions.
s(t) = t⁴/³ / e^t
Verified step by step guidance1
Step 1: Identify the function s(t) = \(\frac{t^{4/3}\)}{e^t}. This is a quotient of two functions, so we will use the Quotient Rule to find the derivative.
Step 2: Recall the Quotient Rule: If you have a function \(\frac{u(t)}{v(t)}\), its derivative is \(\frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2}\). Here, u(t) = t^{4/3} and v(t) = e^t.
Step 3: Differentiate u(t) = t^{4/3}. Use the power rule: \(\frac{d}{dt}\)[t^n] = nt^{n-1}. So, u'(t) = \(\frac{4}{3}\)t^{1/3}.
Step 4: Differentiate v(t) = e^t. The derivative of e^t with respect to t is simply e^t, so v'(t) = e^t.
Step 5: Substitute u(t), u'(t), v(t), and v'(t) into the Quotient Rule formula: \(\frac{\frac{4}{3}\)t^{1/3}e^t - t^{4/3}e^t}{(e^t)^2}. Simplify the expression by factoring and combining like terms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
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Derivatives
Quotient Rule
The Quotient Rule is a formula used to find the derivative of a function that is the ratio of two other functions. If you have a function in the form f(t) = g(t) / h(t), the derivative is given by f'(t) = (g'(t)h(t) - g(t)h'(t)) / (h(t))². This rule is essential for differentiating functions like s(t) = t^(4/3) / e^t.
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Exponential Functions
Exponential functions are functions of the form f(t) = a * e^(kt), where e is the base of the natural logarithm. These functions are characterized by their constant rate of growth or decay, making them crucial in various applications. Understanding how to differentiate exponential functions is vital when applying the Quotient Rule, especially when one of the functions in the ratio is an exponential.
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Related Practice
Textbook Question
Find the slope of the line tangent to the graph of y = sin^−1 x at x=0.
Textbook Question
Find the derivative of the following functions.
y = cos x/sin x + 1
Textbook Question
Find the derivative of the following functions.
y = x² In x
Textbook Question
A surface ship is moving (horizontally) in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle that is 20° below the horizontal. How fast is the submarine’s altitude decreasing?
Textbook Question
63–74. Derivatives of logarithmic functions Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
y = log₈ |tan x|
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