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Find
(dy)/(dx) if
y=5sin x^(2)

Find $\frac{d y}{d x}$ if $y=5 \sin x^{2}$

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Find derivatives using the quotient rule I

Full solution

Q. Find

\frac{d y}{d x}

if

y=5 \sin x^{2}

Identify Function: Let's identify the function we need to differentiate. The function is $y = 5\sin(x^2)$ . We will use the chain rule to differentiate this function because it is a composition of two functions: the sine function and the function $x^2$ .
Apply Chain Rule: 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 $5\sin(u)$ where $u = x^2$ , and the inner function is $x^2$ .
Differentiate Outer Function: First, we differentiate the outer function with respect to $u$ . The derivative of $5\sin(u)$ with respect to $u$ is $5\cos(u)$ , because the derivative of $\sin(u)$ is $\cos(u)$ and we have a constant multiple of $5$ .
Differentiate Inner Function: Next, we differentiate the inner function with respect to $x$ . The derivative of $x^2$ with respect to $x$ is $2x$ , because we apply the power rule which states that the derivative of $x^n$ is $n\cdot x^{(n-1)}$ .
Apply Chain Rule: Now we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us $\frac{dy}{dx} = 5\cos(u) \cdot 2x$ , where $u = x^2$ .
Substitute Back: Substitute $u$ back into the equation to get the derivative in terms of $x$ . This gives us $\frac{dy}{dx} = 5\cos(x^2) \cdot 2x$ .
Simplify Final Answer: Finally, we simplify the expression to get the final answer. $(\frac{dy}{dx} = 10x \cdot \cos(x^2))$ .

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