Use an appropriate form of the chain rule to find dw/dt. w=5 cosx y-sinx z ; x=1 / t, y=t, z=t^3

Name: Problem 9, Chapter 13: Calculus Early Transcendentals
Uploaded: 2020-08-23T22:32:02-08:00
Duration: 2 min 46 s
Channel: James Macpherson
Description: Use an appropriate form of the chain rule to find dw/dt. w=5 cosx y-sinx z ; x=1 / t, y=t, z=t^3

step 1 We're looking for DWDT, which first we'll want to find DWDX times DXDT, which will be negative 5

Use an appropriate form of the chain rule to find dw/dt. w=5...

Key Concepts

Multivariable Chain Rule

The multivariable chain rule is a technique used to differentiate a function that depends on several intermediate variables, each of which is itself a function of another variable. In the context of this problem, the function w depends on x, y, and z, which are all functions of t. To find dw/dt, one takes the partial derivatives of w with respect to each intermediary variable and multiplies these by the corresponding derivatives of the intermediary variables with respect to t, then sums the results.

Partial Derivatives

Partial derivatives are used to measure the rate at which a multivariable function changes with respect to one of its variables while keeping the other variables constant. In the chain rule application for this problem, the partial derivative of w with respect to x, for example, indicates how w changes as x varies (while y and z are held constant), and is a crucial component in forming the full derivative dw/dt.

Differentiation of Trigonometric Functions

Differentiation of trigonometric functions involves using known derivatives such as d/dx (cos x) = -sin x and d/dx (sin x) = cos x. These derivatives are applied within the chain rule framework when the trigonometric functions are composed with expressions involving other variables. In problems like this one, knowing these rules is essential for differentiating the composite trigonometric expressions accurately.

Function Composition

Function composition is the process of substituting one function into another. In the given problem, the variables x, y, and z are expressed as functions of t, and these functions are composed with w. Recognizing and handling the nested structure of these functions is critical in applying the chain rule correctly, as it allows one to break down the differentiation process into manageable steps.

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