Textbook Question
Finding Derivative Values
In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.
f(u) = cot(πu/10), u = g(x) = 5√x, x = 1
3. Techniques of Differentiation
The Chain Rule
8:53 minutesProblem 3.6.52
Textbook Question
In Exercises 41–58, find dy/dt.
y = (1/6)(1 + cos²(7t))³
Verified step by step guidance1
Identify the function y = \( \frac{1}{6} (1 + \cos^2(7t))^3 \) and recognize that you need to find \( \frac{dy}{dt} \). This involves using the chain rule for differentiation.
Apply the chain rule: If \( y = u^3 \) where \( u = 1 + \cos^2(7t) \), then \( \frac{dy}{dt} = 3u^2 \cdot \frac{du}{dt} \).
Differentiate \( u = 1 + \cos^2(7t) \) with respect to t. Use the chain rule again: \( \frac{du}{dt} = 2\cos(7t) \cdot (-\sin(7t)) \cdot 7 \).
Simplify \( \frac{du}{dt} \) to \( -14\cos(7t)\sin(7t) \). This can be further simplified using the double angle identity: \( \sin(2x) = 2\sin(x)\cos(x) \), so \( \frac{du}{dt} = -7\sin(14t) \).
Substitute \( u \) and \( \frac{du}{dt} \) back into the expression for \( \frac{dy}{dt} \): \( \frac{dy}{dt} = \frac{1}{6} \cdot 3(1 + \cos^2(7t))^2 \cdot (-7\sin(14t)) \). Simplify this expression to find the derivative.
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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 differentiation technique used when dealing with composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this problem, the chain rule helps differentiate the nested functions within y = (1/6)(1 + cos²(7t))³, particularly the inner function cos²(7t) and its outer cube.
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Derivative of Trigonometric Functions
Understanding how to differentiate trigonometric functions is crucial for solving this problem. The derivative of cos(x) is -sin(x), and when dealing with cos²(x), the power rule and chain rule are applied. Specifically, for cos²(7t), you need to differentiate using the chain rule, considering the inner function 7t and the outer square.
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Power Rule
The power rule is used to differentiate functions of the form x^n, where the derivative is n*x^(n-1). In this problem, the power rule is applied to the function (1 + cos²(7t))³, where the exponent 3 is brought down, and the function inside is raised to the power of 2, followed by differentiating the inner function using the chain rule.
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