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Evaluate
(d)/(dx)[arctan(2x)] at
x=-2.
Use an exact expression.

Evaluate $\frac{d}{d x}[\arctan (2 x)]$ at $x=-2$ . $\newline$ Use an exact expression.

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Find derivatives of inverse trigonometric functions

Full solution

Q. Evaluate

\frac{d}{d x}[\arctan (2 x)]

at

x=-2

.

\newline

Use an exact expression.

Apply Chain Rule: Use the chain rule to differentiate $\arctan(2x)$ . $f'(x) = \frac{d}{dx}[\arctan(2x)] = \frac{1}{1+(2x)^2} \cdot \frac{d}{dx}(2x)$
Differentiate $2x$ : Differentiate $2x$ with respect to $x$ . $\frac{d}{dx}(2x) = 2$
Substitute Derivative: Substitute the derivative of $2x$ into the chain rule result. $\newline$ $f'(x) = \frac{1}{1+(2x)^2} \times 2$
Simplify Expression: Simplify the expression for the derivative. $f'(x) = \frac{2}{1+4x^2}$
Evaluate at $x = -2$ : Evaluate the derivative at $x = -2$ . $f'(-2) = \frac{2}{1+4(-2)^2}$
Calculate Value: Calculate the value inside the parentheses. $\newline$ $4(-2)^2 = 4\times4 = 16$
Substitute Value: Substitute the calculated value back into the derivative. $f'(-2) = \frac{2}{1+16}$
Add Denominator: Add the values in the denominator. $1+16 = 17$
Complete Calculation: Complete the calculation for the derivative at $x = -2$ . $\newline$ $f'(-2) = \frac{2}{17}$

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