y=sqrt(4x+1) (dy)/(dx)=? Choose 1 answer: (A) 2sqrt(4x+1) (B) (2)/(sqrtx) (c) (2)/(sqrt(4x+1)) (D) (1)/(2sqrt(4x+1))

Identify Function: Identify the function to differentiate. We are given the function y = sqrt(4x+1), and we need to find its derivative with respect to x, which is denoted as (dy)/(dx).Apply Chain Rule: Apply the chain rule for differentiation. 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. In this case, the outer function is the square root function, and the inner function is (4x+1).Differentiate Outer Function: Differentiate the outer function with respect to the inner function. The derivative of sqrt(u) with respect to u is (1)/(2sqrt(u)). Here, u = 4x+1, so the derivative of the outer function is (1)/(2sqrt(4x+1)).Differentiate Inner Function: Differentiate the inner function with respect to x. The derivative of (4x+1) with respect to x is 4, since the derivative of a constant is 0 and the derivative of 4x with respect to x is 4.Multiply Derivatives: Multiply the derivatives of the outer and inner functions. Using the chain rule from Step 2, we multiply the derivative of the outer function from Step 3 by the derivative of the inner function from Step 4. This gives us (dy)/(dx) = (1)/(2sqrt(4x+1)) * 4.Simplify Expression: Simplify the expression. We can simplify the expression by canceling out a factor of 2 from the numerator and denominator. This leaves us with (dy)/(dx) = (2)/(sqrt(4x+1)).

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

y=sqrt(4x+1)

(dy)/(dx)=?
Choose 1 answer:
(A)
2sqrt(4x+1)
(B)
(2)/(sqrtx)
(c)
(2)/(sqrt(4x+1))
(D)
(1)/(2sqrt(4x+1))

$y=\sqrt{4 x+1}$ $\newline$ $\frac{d y}{d x}=?$ $\newline$ Choose $1$ answer: $\newline$ (A) $2 \sqrt{4 x+1}$ $\newline$ (B) $\frac{2}{\sqrt{x}}$ $\newline$ (c) $\frac{2}{\sqrt{4 x+1}}$ $\newline$ (D) $\frac{1}{2 \sqrt{4 x+1}}$

View step-by-step help

Full solution

y=\sqrt{4 x+1}

\newline

\frac{d y}{d x}=?

\newline

Choose

1

answer:

\newline

(A)

2 \sqrt{4 x+1}

\newline

(B)

\frac{2}{\sqrt{x}}

\newline

(c)

\frac{2}{\sqrt{4 x+1}}

\newline

(D)

\frac{1}{2 \sqrt{4 x+1}}

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $y = \sqrt{4x+1}$ , and we need to find its derivative with respect to $x$ , which is denoted as $\frac{dy}{dx}$ .
Apply Chain Rule: Apply the chain rule for differentiation. 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. In this case, the outer function is the square root function, and the inner function is $(4x+1)$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $\sqrt{u}$ with respect to $u$ is $\frac{1}{2\sqrt{u}}$ . Here, $u = 4x+1$ , so the derivative of the outer function is $\frac{1}{2\sqrt{4x+1}}$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $(4x+1)$ with respect to $x$ is $4$ , since the derivative of a constant is $0$ and the derivative of $4x$ with respect to $x$ is $4$ .
Multiply Derivatives: Multiply the derivatives of the outer and inner functions. $\newline$ Using the chain rule from Step $2$ , we multiply the derivative of the outer function from Step $3$ by the derivative of the inner function from Step $4$ . This gives us $\frac{dy}{dx} = \frac{1}{2\sqrt{4x+1}} \times 4$ .
Simplify Expression: Simplify the expression. $\newline$ We can simplify the expression by canceling out a factor of $2$ from the numerator and denominator. This leaves us with $\frac{dy}{dx} = \frac{2}{\sqrt{4x+1}}$ .