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

Identify Terms: Identify the individual terms in the function y = x^2 - csc(x) + 5 that we need to differentiate.Differentiate x^2: Differentiate the first term, x^2, with respect to x. d/dx(x^2) = 2xDifferentiate -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). d/dx(-csc(x)) = d/dx(-1 * csc(x)) = -1 * d/dx(csc(x)) = -1 * (-csc(x)cot(x)) = csc(x)cot(x)Differentiate 5: Differentiate the third term, 5, with respect to x. The derivative of a constant is 0. d/dx(5) = 0Combine Derivatives: Combine the derivatives of the individual terms to find the derivative of the entire function. (dy)/(dx) = d/dx(x^2) + d/dx(-csc(x)) + d/dx(5) (dy)/(dx) = 2x + csc(x)cot(x) + 0

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Q. Find

\frac{dy}{dx}

for the given function.

\newline

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

\newline

\frac{dy}{dx}=

Identify Terms: Identify the individual terms in the function $y = x^2 - \csc(x) + 5$ that we need to differentiate.
Differentiate $x^2$ : Differentiate the first term, $x^2$ , with respect to $x$ . $\frac{d}{dx}(x^2) = 2x$
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 $\csc(x)\cot(x)$ . $\newline$ $\frac{d}{dx}(-\csc(x)) = \frac{d}{dx}(-1 \times \csc(x)) = -1 \times \frac{d}{dx}(\csc(x)) = -1 \times (-\csc(x)\cot(x)) = \csc(x)\cot(x)$
Differentiate $5$ : Differentiate the third term, $5$ , with respect to $x$ . The derivative of a constant is $0$ . $\frac{d}{dx}(5) = 0$
Combine Derivatives: Combine the derivatives of the individual terms to find the derivative of the entire function. $\newline$ $(\frac{dy}{dx}) = \frac{d}{dx}(x^2) + \frac{d}{dx}(-\csc(x)) + \frac{d}{dx}(5)$ $\newline$ $(\frac{dy}{dx}) = 2x + \csc(x)\cot(x) + 0$