How to find the derivative of the function y = tan - 1 ( 3 x ) ?

Question

How to find the derivative of the function       y    =                  tan                  -          1                            (        3        x        )            ?

Hazel Lam · Accepted Answer

Remembering that the derivative of:      y    =                  tan                  -          1                    x       is      y    ′    =          1              1        +                  x          2                    Thus:      y    ′    =          1              1        +                              (            3            x            )                    2                      ⋅    3    =                  3                    1        +        9                  x          2                    .

voteerto55 · Accepted Answer

By the Chain Rule (and knowledge of the derivative of inverse tangent function), the answer is:

\frac{d y}{d x} = \frac{1}{1 + {(3 x)}^{2}} \cdot 3 = \frac{3}{1 + 9 x^{2}}

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If you didn't happen to remember the derivative of the inverse tangent function, you could derive the answer by differentiation of both sides of the equation

\tan (\tan^{- 1} (3 x)) = 3 x

with respect to x to get, by the Chain Rule,

\sec^{2} (\tan^{- 1} (3 x)) \cdot \frac{d}{d x} (\tan^{- 1} (3 x)) = 3

so that

\frac{d}{d x} (\tan^{- 1} (3 x)) = 3 \cos^{2} (\tan^{- 1} (3 x))

.
To simplify this, at this point, you could draw a right triangle and label one of the non-right angles

\tan^{- 1} (3 x)

. Since the tangent of the angle is the length of the opposite side divided by the length of the adjacent side, you could label the opposite side with "3x" and the adjacent side with "1". The Pythagorean Theorem would imply that the length of the hypotenuse is

\sqrt{1 + 9 x^{2}}

. Since the cosine of the angle is the length of the adjacent side divided by the length of the hypotenuse, you'd get

\cos (\tan^{- 1} (3 x)) = \frac{1}{\sqrt{1 + 9 x^{2}}}

.
Thus,

\frac{d}{d x} (\tan^{- 1} (3 x)) = 3 \cos^{2} (\tan^{- 1} (3 x)) = \frac{3}{1 + 9 x^{2}}

How do you find the derivative of the function y=tan^(-1)(3x)?

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