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f(x)=arctan(3x)

f^(')(x)= ?
Choose 1 answer:
(A)
(3)/(1-9x^(2))
(B)
(-3)/(1-9x^(2))
(C)
(3)/(1+9x^(2))
(D)
(-3)/(1+9x^(2))

$f(x)=\arctan (3 x)$ $\newline$ $f^{\prime}(x)=$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{3}{1-9 x^{2}}$ $\newline$ (B) $\frac{-3}{1-9 x^{2}}$ $\newline$ (C) $\frac{3}{1+9 x^{2}}$ $\newline$ (D) $\frac{-3}{1+9 x^{2}}$

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Find derivatives of inverse trigonometric functions

Full solution

Q.

f(x)=\arctan (3 x)

\newline

f^{\prime}(x)=

?

\newline

Choose

1

answer:

\newline

(A)

\frac{3}{1-9 x^{2}}

\newline

(B)

\frac{-3}{1-9 x^{2}}

\newline

(C)

\frac{3}{1+9 x^{2}}

\newline

(D)

\frac{-3}{1+9 x^{2}}

Apply Chain Rule: Use the chain rule to differentiate $f(x) = \arctan(3x)$ . The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Derivative of arctan(u): The outer function is $\text{arctan}(u)$ with $u = 3x$ . The derivative of $\text{arctan}(u)$ with respect to $u$ is $\frac{1}{1+u^2}$ .
Derivative of $3x$ : The inner function is $u = 3x$ . The derivative of $3x$ with respect to $x$ is $3$ .
Apply Chain Rule: Apply the chain rule: $f'(x) = \frac{1}{1+(3x)^2} \cdot \frac{d}{dx}(3x)$ .
Calculate Derivative: Calculate the derivative: $f'(x) = \left(\frac{1}{1+9x^2}\right) \cdot 3$ .
Simplify Expression: Simplify the expression: $f'(x) = \frac{3}{1+9x^2}$ .

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