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s) Find the derivative of
g(x)=-6csc x+3x^(2)sec x

s) Find the derivative of $g(x)=-6 \csc x+3 x^{2} \sec x$

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Find derivatives using the quotient rule II

Full solution

Q. s) Find the derivative of

g(x)=-6 \csc x+3 x^{2} \sec x

Differentiate $-6\csc(x)$ : Differentiate the first term $-6\csc(x)$ . The derivative of $\csc(x)$ is $-\csc(x)\cot(x)$ , so the derivative of $-6\csc(x)$ is $6\csc(x)\cot(x)$ .
Differentiate $3x^{2}\sec(x)$ : Differentiate the second term $3x^{2}\sec(x)$ . We will use the product rule for differentiation, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Let $u = x^2$ and $v = \sec(x)$ , then $u' = 2x$ and $v' = \sec(x)\tan(x)$ . The derivative of $3x^{2}\sec(x)$ is $3(u'v + uv')$ .
Apply product rule for differentiation: Calculate the derivative of $3x^{2}\sec(x)$ using the product rule. $\newline$ Substitute $u$ , $u'$ , $v$ , and $v'$ into the product rule formula: $\newline$ $3(u'v + uv') = 3(2x\cdot\sec(x) + x^2\cdot\sec(x)\tan(x))$ . $\newline$ Simplify the expression: $\newline$ $3(u'v + uv') = 3(2x\cdot\sec(x) + x^2\cdot\sec(x)\tan(x)) = 6x\cdot\sec(x) + 3x^2\cdot\sec(x)\tan(x)$ .
Calculate derivative using product rule: Combine the derivatives of both terms to find the derivative of $g(x)$ . The derivative of $g(x)$ is the sum of the derivatives of its terms: $g'(x) = 6\csc(x)\cot(x) + 6x\cdot\sec(x) + 3x^2\cdot\sec(x)\tan(x)$ .

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