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Chapter 3: Problem 33

Find the derivative of the following functions. $$y=\sec x+\csc x$$

Short Answer

Expert verified
Answer: The derivative of the function \(y=\sec x + \csc x\) is \(\frac{dy}{dx} = \sec x \tan x - \csc x \cot x\).

Step by step solution

01

Identify the functions to derive

We are given the function \(y=\sec x+\csc x\). To find the derivative, we need to differentiate both the secant function \(\sec x\) and the cosecant function \(\csc x\) individually.
02

Derivative of the secant function

Recall the derivative of the secant function \(\sec x\): $$\frac{d}{dx}(\sec x)=\sec x \tan x$$
03

Derivative of the cosecant function

Recall the derivative of the cosecant function \(\csc x\): $$\frac{d}{dx}(\csc x)=-\csc x \cot x$$
04

Apply the derivatives to the function

Using the individual derivatives from Steps 2 and 3, differentiate the given function \(y=\sec x+\csc x\): \begin{align*} \frac{dy}{dx} &= \frac{d}{dx}(\sec x)+ \frac{d}{dx}(\csc x) \\ &= \sec x \tan x -\csc x \cot x \end{align*} The derivative of the function \(y=\sec x+\csc x\) is: $$\frac{dy}{dx} = \sec x \tan x -\csc x \cot x$$

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Most popular questions from this chapter

Compute the derivative of the following functions. \(h(x)=\frac{(x+1)}{x^{2} e^{x}}\)

Use any method to evaluate the derivative of the following functions. \(y=\frac{x-a}{\sqrt{x}-\sqrt{a}} ; a\) is a positive constant.

Use a trigonometric identity to show that the derivatives of the inverse cotangent and inverse cosecant differ from the derivatives of the inverse tangent and inverse secant, respectively, by a multiplicative factor of -1

A challenging second derivative Find \(\frac{d^{2} y}{d x^{2}},\) where \(\sqrt{y}+x y=1\).

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}(f(x) g(x))\right|_{x=1}$$
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