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Chapter 3: Problem 32

Find the derivative of the following functions. $$y=\tan x+\cot x$$

Short Answer

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Question: Find the derivative of the function \(y = \tan x + \cot x\). Solution: The derivative of the given function is \(y' = \sec^2 x - \csc^2 x\).

Step by step solution

01

Differentiate \(\tan x\)

To find the derivative of \(\tan x\), we can use the derivative rule for trigonometric functions, which states that the derivative of \(\tan x\) is given by the function \(\sec^2 x\). Therefore, $$\frac{d}{dx}(\tan x) = \sec^2 x$$.
02

Differentiate \(\cot x\)

To find the derivative of \(\cot x\), we can use the derivative rule for trigonometric functions, which states that the derivative of \(\cot x\) is given by the function \(-\csc^2 x\). Therefore, $$\frac{d}{dx}(\cot x) = -\csc^2 x$$.
03

Combine the derivatives

Now that we have the derivatives of the individual functions, we will find the derivative of the sum by using the sum rule of differentiation. The sum rule states that the derivative of the sum of two functions is the sum of their derivatives, which means, $$\frac{d}{dx}(y) = \frac{d}{dx}(\tan x) + \frac{d}{dx}(\cot x)$$. Substitute the results from steps 1 and 2, $$\frac{d}{dx}(y) = \sec^2 x - \csc^2 x$$. The derivative of the function \(y = \tan x + \cot x\) is given by, $$y' = \sec^2 x - \csc^2 x$$.

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