Find the derivative of y with respect to the appropriate variable. y=(1-t) coth^-1 √(t)

Name: Problem 29, Chapter 7: University Calculus: Early Transcendentals
Uploaded: 2020-04-15T13:48:33-08:00
Duration: 2 min 9 s
Channel: Matthew Allcock
Description: Find the derivative of y with respect to the appropriate variable. y=(1-t) coth^-1 √(t)

step 1 In this video, we're going to go through the answer to question number 29 from chapter 7 .3. So

Find the derivative of y with respect to the appropriate...

Key Concepts

Product Rule

The product rule is a key differentiation tool used when finding the derivative of a product of two functions. It states that if you have a function y = u(t) * v(t), then the derivative y' is given by u'(t) * v(t) + u(t) * v'(t). This rule is essential in differentiating complex expressions where functions of the variable multiply each other.

Chain Rule

The chain rule is used to differentiate composite functions, i.e., functions nested within other functions. It states that the derivative of a composite function f(g(t)) is f'(g(t)) * g'(t). This concept is particularly important when differentiating functions where the argument of one function is itself another function of the variable.

Derivative of Inverse Hyperbolic Functions

When differentiating inverse hyperbolic functions, such as the inverse hyperbolic cotangent (coth?¹), special formulas are employed. For instance, the derivative of coth?¹(x) with respect to x is typically given by -1/(1 - x²), with the appropriate domain restrictions. This concept is crucial when the inverse hyperbolic function appears as part of a more complex expression.

Differentiation of the Square Root Function

The square root function, expressed as ?t or t^(1/2), is a common elementary function encountered in calculus. Its derivative is found using the power rule, which leads to d(?t)/dt = 1/(2?t). This fundamental concept supports the differentiation of composite functions where a square root appears as the inner function.

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