Question

Find the derivative. $H(\phi)=(\cot \phi+\csc \phi)(\tan \phi-\sin \phi)$

   Find the derivative.
$H(\phi)=(\cot \phi+\csc \phi)(\tan \phi-\sin \phi)$

Calculus

Earl W. Swokowski 5th Edition

Chapter 3, Problem 27

Instant Answer

Solved by Expert Steven Clarke

03/01/2022

Step 1

We know that $\cot \phi = \frac{1}{\tan \phi}$ and $\csc \phi = \frac{1}{\sin \phi}$. So, we can rewrite the function as follows: $$ H(\phi)=(\frac{1}{\tan \phi}+\frac{1}{\sin \phi})(\tan \phi-\sin \phi) $$ Show more…

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Find the derivative. $H(\phi)=(\cot \phi+\csc \phi)(\tan \phi-\sin \phi)$

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Key Concepts

Algebraic Simplification

Algebraic simplification involves manipulating and reducing expressions to their simplest form before or after applying calculus techniques. Simplification can make differentiation easier and the resulting derivative more understandable by reducing complexity and canceling unnecessary terms.

Trigonometric Functions and Their Derivatives

Understanding the derivatives of standard trigonometric functions, such as sine, cosine, tangent, cotangent, secant, and cosecant, is crucial in calculus. These derivatives are used in problems involving trigonometric expressions and are vital for applying differentiation rules correctly when such functions are involved.

Differentiation

Differentiation is the process in calculus 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 fundamental concept used to analyze the behavior of functions, including finding slopes of tangent lines, rates of change, and optimizing variables.

Product Rule

The product rule is a technique used to differentiate functions that are the product of two or more simpler functions. It states that the derivative of a product is the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function, and is essential when handling functions expressed as products.

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