y=arctan(-4x) Evaluate (dy)/(dx) at x=3. Use an exact expression.

Differentiate using chain rule: step_1: Differentiate y = arctan(-4x) with respect to x using the chain rule. The derivative of arctan(u) with respect to u is 1/(1+u^2), so by the chain rule, dy/dx = (1/(1+(-4x)^2)) * d(-4x)/dx.Calculate derivative of -4x: step_2: Calculate the derivative of -4x with respect to x. d(-4x)/dx = -4.Substitute derivative into result: step_3: Substitute the derivative of -4x into the result from step 1. dy/dx = (1/(1+(-4x)^2)) * (-4).Simplify expression for dy/dx: step_4: Simplify the expression for dy/dx. dy/dx = -4/(1+16x^2).Evaluate dy/dx at x = 3: step_5: Evaluate dy/dx at x = 3. dy/dx at x = 3 is -4/(1+16(3)^2).Calculate exact expression at x = 3: step_6: Calculate the exact expression for dy/dx at x = 3. dy/dx at x = 3 is -4/(1+16*9).Simplify the denominator: step_7: Simplify the denominator. 1+16*9 = 1+144.Add numbers in the denominator: step_8: Add the numbers in the denominator. 1+144 = 145.Write final expression for dy/dx: step_9: Write the final expression for dy/dx at x = 3. dy/dx at x = 3 is -4/145.

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y=arctan(-4x)
Evaluate
(dy)/(dx) at
x=3.
Use an exact expression.

$y=\arctan (-4 x)$ $\newline$ Evaluate $\frac{d y}{d x}$ at $x=3$ . $\newline$ Use an exact expression.

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Calculus

Find derivatives of inverse trigonometric functions

Full solution

y=\arctan (-4 x)

\newline

Evaluate

\frac{d y}{d x}

x=3

\newline

Use an exact expression.

Differentiate using chain rule: step_1: Differentiate $y = \arctan(-4x)$ with respect to $x$ using the chain rule. $\newline$ The derivative of $\arctan(u)$ with respect to $u$ is $\frac{1}{1+u^2}$ , so by the chain rule, $\frac{dy}{dx} = \left(\frac{1}{1+(-4x)^2}\right) * \frac{d(-4x)}{dx}$ .
Calculate derivative of $-4x$ : step_2: Calculate the derivative of $-4x$ with respect to $x$ . $\frac{d(-4x)}{dx} = -4.$
Substitute derivative into result: step_3: Substitute the derivative of $-4x$ into the result from step $1$ . $\newline$ \frac{dy}{dx} = \left(\frac{1}{1+(\-4x)^2}\right) * (\-4).
Simplify expression for $\frac{dy}{dx}$ : step_4: Simplify the expression for $\frac{dy}{dx}$ . $\frac{dy}{dx} = \frac{-4}{1+16x^2}$ .
Evaluate $\frac{dy}{dx}$ at $x = 3$ : step_5: Evaluate $\frac{dy}{dx}$ at $x = 3$ . $\frac{dy}{dx}$ at $x = 3$ is $-\frac{4}{1+16(3)^2}$ .
Calculate exact expression at $x = 3$ : step_6: Calculate the exact expression for $\frac{dy}{dx}$ at $x = 3$ . $\frac{dy}{dx}$ at $x = 3$ is $-\frac{4}{1+16\cdot 9}$ .
Simplify the denominator: step_7: Simplify the denominator. $\newline$ $1+16\times 9 = 1+144$ .
Add numbers in the denominator: step_8: Add the numbers in the denominator. $1+144 = 145$ .
Write final expression for $\frac{dy}{dx}$ : step_9: Write the final expression for $\frac{dy}{dx}$ at $x = 3$ . $\newline$ $\frac{dy}{dx}$ at $x = 3$ is $-\frac{4}{145}$ .