Textbook Question
Find ds/dt when θ = 3π/2 if s = cosθ and dθ/dt = 5.
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.6.10
In Exercises 9–18, write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.
y = (4 − 3x)⁹
Verified step by step guidance1
Step 1: Identify the inner function u = g(x). In this case, the expression inside the parentheses is the inner function, so we have u = 4 - 3x.
Step 2: Express the original function y in terms of u. Since y = (4 - 3x)⁹, we can rewrite it as y = f(u) = u⁹.
Step 3: Differentiate y = f(u) with respect to u. Using the power rule, the derivative of u⁹ with respect to u is dy/du = 9u⁸.
Step 4: Differentiate u = g(x) with respect to x. The derivative of u = 4 - 3x with respect to x is du/dx = -3.
Step 5: Apply the chain rule to find dy/dx. The chain rule states that dy/dx = (dy/du) * (du/dx). Substitute the derivatives found in steps 3 and 4: dy/dx = 9u⁸ * (-3). Finally, substitute u = 4 - 3x back into the expression to get dy/dx as a function of x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chain Rule
The Chain Rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative dy/dx can be found by multiplying the derivative of y with respect to u (dy/du) by the derivative of u with respect to x (du/dx). This allows for the differentiation of complex functions by breaking them down into simpler parts.
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Function Composition
Function composition involves creating a new function by combining two functions, where the output of one function becomes the input of another. In the context of the given problem, we express y as a function of u, and u as a function of x. Understanding how to manipulate and express these relationships is crucial for applying the Chain Rule effectively and finding derivatives.
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Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where y is not explicitly solved for x. In cases where y is defined in terms of another variable u, and u in terms of x, implicit differentiation allows us to find dy/dx without isolating y. This method is particularly useful when dealing with complex functions or when the relationship between variables is not straightforward.
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Related Practice
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
______
𝓻 = √2θ sinθ
Textbook Question
In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.
g(x) = 8 / x², (2, 2)
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x² sin² (2x²)
Textbook Question
Derivatives in Differential Form
In Exercises 17–28, find dy.
2y³/² + xy − x = 0
Textbook Question
Find the derivatives of all orders of the functions in Exercises 29–32.
y = x⁵ / 120