Textbook Question
5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = (5x²+11x)^4/3
3. Techniques of Differentiation
The Chain Rule
5:33 minutesProblem 42
Textbook Question
Calculate the derivative of the following functions.
y = cos4 θ + sin4 θ
Verified step by step guidance1
Step 1: Recognize that the function y = \cos^4 \theta + \sin^4 \theta is a sum of two terms, each raised to the fourth power. We will need to use the chain rule to differentiate each term separately.
Step 2: Apply the chain rule to the first term \cos^4 \theta. The chain rule states that if you have a function u(\theta) raised to a power n, the derivative is n * u(\theta)^{n-1} * u'(\theta). Here, u(\theta) = \cos \theta and n = 4.
Step 3: Differentiate \cos \theta with respect to \theta to find u'(\theta). The derivative of \cos \theta is -\sin \theta.
Step 4: Combine the results from Steps 2 and 3 to find the derivative of \cos^4 \theta. It will be 4 * \cos^3 \theta * (-\sin \theta).
Step 5: Repeat Steps 2-4 for the second term \sin^4 \theta. The derivative of \sin \theta is \cos \theta, so the derivative of \sin^4 \theta is 4 * \sin^3 \theta * \cos \theta. Finally, add the derivatives of both terms to get the derivative of the entire function.
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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 value changes. It is a fundamental concept in calculus that represents the slope of the tangent line to the curve of the function at any given point. The derivative can be calculated using various rules, such as the power rule, product rule, and chain rule.
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Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is the composition of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions raised to a power, as seen in the given problem.
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Trigonometric Functions
Trigonometric functions, such as sine and cosine, are fundamental functions in calculus that relate angles to ratios of sides in right triangles. They are periodic functions and have specific derivatives: the derivative of sin(θ) is cos(θ), and the derivative of cos(θ) is -sin(θ). Understanding these derivatives is essential for calculating the derivative of functions involving trigonometric expressions.
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