If w=sin(x y z), x=1-3 t, y=e^1-t and z=4 t, find (∂w)/(∂t)

Name: Problem 247, Chapter 4: Calculus Volume 3
Uploaded: 2020-10-16T04:23:22-08:00
Duration: 3 min 49 s
Channel: Victoria Dollar
Description: If w=sin(x y z), x=1-3 t, y=e^1-t and z=4 t, find (∂w)/(∂t)

step 1 Okay, so for this problem, we have W equals sign of X, Y, Z. I'm given that X equals 1 minus 3T,

If w=sin(x y z), x=1-3 t, y=e^1-t and z=4 t, find...

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus for differentiating composite functions. It states that when one function is applied to the result of another, the derivative of the entire function with respect to an independent variable is the derivative of the outer function with respect to the inner function times the derivative of the inner function with respect to that variable.

Composite Functions in Multivariable Calculus

In multivariable calculus, functions often depend on several intermediate variables, each of which may itself be a function of an independent parameter. The multivariable chain rule extends the basic chain rule by taking into account the derivatives of all the intermediary functions, ensuring that the overall rate of change with respect to the parameter is correctly computed.

Differentiation of Transcendental Functions

Transcendental functions, such as the sine function, have specific derivative rules that are applied when they are part of a composite function. When a transcendental function is combined with other functions (via composition), the standard differentiation rules must be modified using the chain rule to incorporate the derivatives of the inner functions.

Parameterization

Parameterization involves expressing variables as functions of an independent parameter. In problems where multiple variables are defined in terms of a single parameter, it becomes necessary to differentiate each variable with respect to that parameter and then use the chain rule to find the overall derivative of a function that depends on these variables.

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