Question

find $d y$. $y=2 \cot \left(\frac{1}{\sqrt{x}}\right)$

   find $d y$.
$y=2 \cot \left(\frac{1}{\sqrt{x}}\right)$

Thomas Calculus

Thomas Calculus

George B. Thomas Jr. 14th Edition

Chapter 3, Problem 28

Instant Answer

Solved by Expert Matt Just

05/09/2019

Step 1

We know that $y=2 \cot \left(\frac{1}{\sqrt{x}}\right)$, so we can use the chain rule to find the derivative. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Show more…

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find $d y$. $y=2 \cot \left(\frac{1}{\sqrt{x}}\right)$

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Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function’s value changes with respect to changes in its input. It is a foundational technique in calculus that provides insights into the behavior and rate of change of functions.

Chain Rule

The chain rule is a method used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function, evaluated at the inner function, multiplied by the derivative of the inner function. This rule is crucial when dealing with nested functions.

Trigonometric Derivatives

Trigonometric derivatives involve finding the derivatives of trigonometric functions using established formulas. For example, the derivative of the cotangent function is -cosecant squared, a formula that is essential when differentiating expressions involving trigonometric functions.

Function Composition

Function composition refers to the application of one function to the results of another. Understanding composition is key to applying the chain rule, as it requires determining the derivative of the outer function and then multiplying by the derivative of the inner function.

Reciprocal and Power Rule

The reciprocal and power rules are techniques used to differentiate functions involving negative exponents or reciprocal terms. For instance, expressions like x^(-1/2) are differentiated by applying the power rule, which simplifies the process of finding derivatives of functions expressed in fractional or reciprocal forms.

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