For , let . Determine…

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

For x>0x>0, let f(x)=cot⁡ ⁣(5x2+7)f(x)=\cot\!\bigl(\sqrt{5x^{2}+7}\bigr). Determine f′(x)f'(x). (a) 5x5x2+7csc⁡2 ⁣(5x2+7)\dfrac{5x}{\sqrt{5x^{2}+7}}\csc^{2}\!\bigl(\sqrt{5x^{2}+7}\bigr) (b) −5x5x2+7sec⁡2 ⁣(5x2+7)-\dfrac{5x}{\sqrt{5x^{2}+7}}\sec^{2}\!\bigl(\sqrt{5x^{2}+7}\bigr) (c) −5x5x2+7csc⁡2 ⁣(5x2+7)-\dfrac{5x}{\sqrt{5x^{2}+7}}\csc^{2}\!\bigl(\sqrt{5x^{2}+7}\bigr) (d) −5x csc⁡2 ⁣(5x2+7)-5x\,\csc^{2}\!\bigl(\sqrt{5x^{2}+7}\bigr)

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