Derivative of a product of trig functions: 3sin⁡(x)cot⁡(x) −3cos⁡xsin2x

You need to use the product rule: (3sin⁡xcot⁡x)′= 3(sin⁡x)′cot⁡x+3sin⁡x(cot⁡x)′= 3cos⁡xcot⁡x+3sin⁡x(−csc2x)= 3cos⁡xcot⁡x−3sin⁡xcsc2x= 3cos⁡xcos⁡xsin⁡x−3sin⁡x1sin2x= 3cos2xsin⁡x−31sin⁡x= 31sin⁡x(cos2x−1)= 31sin⁡x(1−sin2x−1)=−3sin⁡x A much simpler way to do this, as was mentioned in the comments section, is to notice that 3sin⁡xcot⁡x=3sin⁡xcos⁡xsin⁡x=3cos⁡x which is trivial to differentiate: (3cos⁡x)′=3(cos⁡x)′=3(−sin⁡x)=−3sin⁡x

"Derivative of a product of trig functions: 3sin⁡(x)cot⁡(x)..." was answered by our experts

Arely Briggs

Answered question

2022-01-24

Derivative of a product of trig functions:

3 \sin (x) \cot (x)

\frac{- 3 \cos x}{\sin^{2} x}

Answer & Explanation

Maritza Mccall

Beginner2022-01-25Added 17 answers

You need to use the product rule:

(3 \sin x \cot x)' =

3 (\sin x)' \cot x + 3 \sin x (\cot x)' =

3 \cos x \cot x + 3 \sin x (- \csc^{2} x) =

3 \cos x \cot x - 3 \sin x \csc^{2} x =

3 \cos x \cos x \sin x - 3 \sin x \frac{1}{\sin^{2} x} =

3 \frac{\cos^{2} x}{\sin x} - 3 \frac{1}{\sin x} =

3 \frac{1}{\sin x} (\cos^{2} x - 1) =

3 \frac{1}{\sin x} (1 - \sin^{2} x - 1) = - 3 \sin x

A much simpler way to do this, as was mentioned in the comments section, is to notice that

3 \sin x \cot x = 3 \sin x \frac{\cos x}{\sin x} = 3 \cos x

which is trivial to differentiate:

(3 \cos x)' = 3 (\cos x)' = 3 (- \sin x) = - 3 \sin x

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Calculus

What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders?
How do I find the y-intercept of a parabola?
What are the vertices of $9 x^{2} + 16 y^{2} = 144$ ?
How to determine the rate of change of a function?
Why are the tangents for 90 and 270 degrees undefined?
How to find the center and radius of the circle $x^{2} + y^{2} - 6 x + 8 y = 0$ ?
What is multiplicative inverse of a number?
How to find the continuity of a function on a closed interval?
How do I find the tangent line of a function?
How to find vertical asymptotes using limits?
How to find the center and radius of the circle $x^{2} - 12 x + y^{2} + 4 y + 15 = 0$ ?
Let f be a function so that (below). Which must be true?
I. f is continuous at x=2
II. f is differentiable at x=2
III. The derivative of f is continuous at x=2
(A) I (B) II (C) I and II (D) I and III (E) II and III
How to find the center and radius of the circle given $x^{2} + y^{2} + 8 x - 6 y = 0$ ?
How to find the center and radius of the circle $x^{2} + y^{2} + 4 x - 8 y + 4 = 0$ ?
How to identify the center and radius of the circle ${(x + 3)}^{2} + {(y - 8)}^{2} = 16$ ?

Didn't find what you were looking for?