How To Find The Derivative Of Cot(X) Using Quotient Rule: Step-By-Step Guide

Derivative of cot(x)

-csc^2u du

The derivative of cot(x) can be found using the quotient rule. Recall that:

(1) The quotient rule states that if f(x) and g(x) are differentiable functions, then the derivative of f(x)/g(x) is given by [(g(x)*f'(x)) – (f(x)*g'(x))]/[g(x)]^2.

Now, let’s apply the quotient rule to find the derivative of cot(x):

Let f(x) = 1 and g(x) = tan(x). Then, using the trigonometric identity cot(x) = cos(x)/sin(x), we can write cot(x) as:

cot(x) = cos(x)/sin(x) = f(x)/g(x)

Now, we need to find f'(x) and g'(x):

f'(x) = 0, since the derivative of a constant is 0.

g'(x) = sec^2(x), since the derivative of tan(x) is sec^2(x).

Using the quotient rule with these values, we get:

cot'(x) = [(sin(x)*0) – (cos(x)*sec^2(x))]/[sin(x)]^2

Simplifying this expression, we get:

cot'(x) = -cos(x)*csc^2(x)

Therefore, the derivative of cot(x) is -cos(x)*csc^2(x).

More Answers: