Math  /  Calculus

QuestionGiven the function y=4sin(3x)y=4 \sin (3 \sqrt{x}), find dydx\frac{d y}{d x}.
Answer Attempt 1out of 2 dydx=\frac{d y}{d x}= \square Submit Answe

Studdy Solution

STEP 1

1. We are given the function y=4sin(3x) y = 4 \sin (3 \sqrt{x}) .
2. We need to find the derivative dydx\frac{d y}{d x}.

STEP 2

1. Identify the composition of functions involved in y=4sin(3x) y = 4 \sin (3 \sqrt{x}) .
2. Apply the chain rule to differentiate the composite function.
3. Differentiate the outer function.
4. Differentiate the inner function.
5. Combine the derivatives to find dydx\frac{d y}{d x}.

STEP 3

Identify the composition of functions: - The outer function is u=4sin(u) u = 4 \sin(u) where u=3x u = 3 \sqrt{x} . - The inner function is u=3x u = 3 \sqrt{x} .

STEP 4

Apply the chain rule: The chain rule states that if y=f(g(x)) y = f(g(x)) , then dydx=f(g(x))g(x)\frac{d y}{d x} = f'(g(x)) \cdot g'(x).

STEP 5

Differentiate the outer function 4sin(u) 4 \sin(u) with respect to u u :
ddu[4sin(u)]=4cos(u) \frac{d}{du} [4 \sin(u)] = 4 \cos(u)

STEP 6

Differentiate the inner function u=3x u = 3 \sqrt{x} with respect to x x :
First, rewrite u=3x u = 3 \sqrt{x} as u=3x1/2 u = 3x^{1/2} .
ddx[3x1/2]=312x1/2=32x1/2 \frac{d}{dx} [3x^{1/2}] = 3 \cdot \frac{1}{2} x^{-1/2} = \frac{3}{2} x^{-1/2}

STEP 7

Combine the derivatives using the chain rule:
dydx=4cos(3x)32x1/2 \frac{d y}{d x} = 4 \cos(3 \sqrt{x}) \cdot \frac{3}{2} x^{-1/2}
Simplify the expression:
dydx=6cos(3x)x1/2 \frac{d y}{d x} = 6 \cos(3 \sqrt{x}) \cdot x^{-1/2}
The derivative dydx\frac{d y}{d x} is:
6cos(3x)x1/2 \boxed{6 \cos(3 \sqrt{x}) \cdot x^{-1/2}}

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