hw12 derivatives of trigonometric functions (target c3; §3.5) score: 5/7 answered: 5/7 question 6 if (…

hw12 derivatives of trigonometric functions (target c3; §3.5) score: 5/7 answered: 5/7 question 6 if ( f(x)=(4x^{2}-2x + 1)cot(x)), find ( f(x)=) question help: message instructor submit question

hw12 derivatives of trigonometric functions (target c3; §3.5) score: 5/7 answered: 5/7 question 6 if ( f(x)=(4x^{2}-2x + 1)cot(x)), find ( f(x)=) question help: message instructor submit question

Answer

Explanation:

Step1: Apply product - rule

The product - rule states that if (y = u\cdot v), then (y'=u'v + uv'). Let (u = 4x^{2}-2x + 1) and (v=\cot(x)). First, find (u') and (v'). (u'=\frac{d}{dx}(4x^{2}-2x + 1)=8x-2) (v'=\frac{d}{dx}(\cot(x))=-\csc^{2}(x))

Step2: Calculate (f'(x))

Using the product - rule (f'(x)=u'v+uv'), we substitute (u), (u'), (v), and (v') into the formula. (f'(x)=(8x - 2)\cot(x)+(4x^{2}-2x + 1)(-\csc^{2}(x))=(8x - 2)\cot(x)-(4x^{2}-2x + 1)\csc^{2}(x))

Answer:

((8x - 2)\cot(x)-(4x^{2}-2x + 1)\csc^{2}(x))