Get the answer to "What is the derivative of y=arctan⁡(x) ?"

expeditiupc

Answered question

2021-12-31

What is the derivative of

y = \arctan (x)

Mary Nicholson

Beginner2022-01-01Added 38 answers

The derivative of

y = \arctan x

y^{'} = \frac{1}{1 + x^{2}}

We can derive this by using implicit differentiation.
Since inverse tangent is hard to deal with, we rewrite it as

\tan (y) = x

By implicitly differentiating with respect to x,

\sec^{2} (y) \cdot y^{'} = 1

By solbing for y' and using

\sec^{2} (y) = 1 + \tan^{2} (y)

y^{'} = \frac{1}{\sec^{2} (y)} = \frac{1}{\tan^{2} (y)}

Hence,

y^{'} = \frac{1}{1 + x^{2}}

alkaholikd9

Beginner2022-01-02Added 37 answers

Step 1: Rearrange

y = \arctan (x)

\tan (y) = x

.
Step 2: Use implicit differentiation to differentiate this with respect to x, which gives us:

(\frac{d y}{d x}) \cdot {(\sec (y))}^{2} = 1

Step 3: Rearrange this equation to give us:

\frac{d y}{d x} = \frac{1}{\sec^{2} (y)}

nick1337

Expert2022-01-08Added 777 answers

This helped.

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