Math / Calculus

Question11. What is the slope of the line tangent to the curve $y=\arctan (4 x)$ at the point at which $x=\frac{1}{4}$ ? (A) 2 (B) $\frac{1}{2}$ (C) 0 (D) $-\frac{1}{2}$ (E) -2 $\frac{1}{4} \cdot \frac{1}{4}$ $\frac{5}{4} \quad \frac{1}{16}$

Studdy Solution

STEP 1

1. The function given is $y = \arctan(4x)$ .
2. We need to find the slope of the tangent line at $x = \frac{1}{4}$ .

STEP 2

1. Differentiate the function to find the derivative.
2. Evaluate the derivative at $x = \frac{1}{4}$ .
3. Identify the slope of the tangent line.

STEP 3

Differentiate the function $y = \arctan(4x)$ with respect to $x$ .
The derivative of $y = \arctan(u)$ with respect to $x$ is:
$\frac{dy}{dx} = \frac{1}{1 + u^2} \cdot \frac{du}{dx}$
For $u = 4x$ , we have:
$\frac{du}{dx} = 4$
Thus, the derivative is:
$\frac{dy}{dx} = \frac{1}{1 + (4x)^2} \cdot 4$
$= \frac{4}{1 + 16x^2}$

STEP 4

Evaluate the derivative at $x = \frac{1}{4}$ .
Substitute $x = \frac{1}{4}$ into the derivative:
$\frac{dy}{dx} = \frac{4}{1 + 16\left(\frac{1}{4}\right)^2}$
$= \frac{4}{1 + 16 \cdot \frac{1}{16}}$
$= \frac{4}{1 + 1}$
$= \frac{4}{2}$
$= 2$

STEP 5

Identify the slope of the tangent line.
The slope of the tangent line at $x = \frac{1}{4}$ is $\boxed{2}$ .
The correct answer is (A) 2.

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