Find (dy)/(dx) for the given function. dy_=◻=x^(2)-csc(x)+5

Identify Function: Identify the function to differentiate. We are given the function y = x^2 - csc(x) + 5 and we need to find its derivative with respect to x.Apply Sum Rule: Apply the sum rule for differentiation. The sum rule states that the derivative of a sum of functions is the sum of their derivatives. Therefore, we can differentiate each term of y separately.Differentiate x^2: Differentiate the first term, x^2, with respect to x. Using the power rule, which states that the derivative of x^n with respect to x is n*x^(n-1), we find that the derivative of x^2 is 2*x^(2-1) = 2x.Differentiate -csc(x): Differentiate the second term, -csc(x), with respect to x. The derivative of csc(x) is -csc(x)cot(x), so the derivative of -csc(x) is csc(x)cot(x).Differentiate 5: Differentiate the third term, 5, with respect to x. The derivative of a constant is 0, so the derivative of 5 with respect to x is 0.Combine Derivatives: Combine the derivatives of the individual terms. The derivative of y with respect to x is the sum of the derivatives of the individual terms, which is 2x + csc(x)cot(x) + 0.Simplify Derivative: Simplify the derivative. Since the derivative of the constant term is 0, it does not contribute to the final expression. The simplified derivative of y with respect to x is dy/dx = 2x + csc(x)cot(x).

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Find $\frac{dy}{dx}$ for the given function. $\newline$ $dy_=\square=x^{2}-\csc(x)+5$

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Algebra 2

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Q. Find

\frac{dy}{dx}

for the given function.

\newline

dy_=\square=x^{2}-\csc(x)+5

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $y = x^2 - \csc(x) + 5$ and we need to find its derivative with respect to $x$ .
Apply Sum Rule: Apply the sum rule for differentiation. $\newline$ The sum rule states that the derivative of a sum of functions is the sum of their derivatives. Therefore, we can differentiate each term of $y$ separately.
Differentiate $x^2$ : Differentiate the first term, $x^2$ , with respect to $x$ . Using the power rule, which states that the derivative of $x^n$ with respect to $x$ is $n\cdot x^{(n-1)}$ , we find that the derivative of $x^2$ is $2\cdot x^{(2-1)} = 2x$ .
Differentiate $-\csc(x)$ : Differentiate the second term, $-\csc(x)$ , with respect to $x$ . The derivative of $\csc(x)$ is $-\csc(x)\cot(x)$ , so the derivative of $-\csc(x)$ is $\csc(x)\cot(x)$ .
Differentiate $5$ : Differentiate the third term, $5$ , with respect to $x$ . The derivative of a constant is $0$ , so the derivative of $5$ with respect to $x$ is $0$ .
Combine Derivatives: Combine the derivatives of the individual terms. $\newline$ The derivative of $y$ with respect to $x$ is the sum of the derivatives of the individual terms, which is $2x + \csc(x)\cot(x) + 0$ .
Simplify Derivative: Simplify the derivative. $\newline$ Since the derivative of the constant term is $0$ , it does not contribute to the final expression. The simplified derivative of $y$ with respect to $x$ is $\frac{dy}{dx} = 2x + \csc(x)\cot(x)$ .