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Given
y=-2sin(2x), find
(dy)/(dx).
Answer:
(dy)/(dx)=

Given $y=-2 \sin (2 x)$ , find $\frac{d y}{d x}$ . $\newline$ Answer: $\frac{d y}{d x}=$

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Evaluate a nonlinear function

Full solution

Q. Given

y=-2 \sin (2 x)

, find

\frac{d y}{d x}

.

\newline

Answer:

\frac{d y}{d x}=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $y = -2\sin(2x)$ and we need to find its derivative with respect to $x$ .
Apply Chain Rule: Apply the chain rule for differentiation. $\newline$ 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 $-2\sin(u)$ where $u = 2x$ , and the inner function is $2x$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function $u$ . The derivative of $-2\sin(u)$ with respect to $u$ is $-2\cos(u)$ , because the derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$ and we have a constant multiplier of $-2$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $2x$ with respect to $x$ is $2$ , because the derivative of $x$ with respect to $x$ is $1$ and we have a constant multiplier of $2$ .
Apply Chain Rule: Apply the chain rule by multiplying the derivatives from Step $3$ and Step $4$ . $\newline$ $(\frac{dy}{dx}) = (-2\cos(u)) \times (2)$ where $u = 2x$ .
Substitute Back: Substitute $u$ back into the equation to express the derivative in terms of $x$ . $\frac{dy}{dx} = (-2\cos(2x)) \times (2) = -4\cos(2x)$ .

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