d/dx cotx
To find the derivative of cot(x), we can use the derivative formulas for trigonometric functions
To find the derivative of cot(x), we can use the derivative formulas for trigonometric functions. Let’s walk through the process step-by-step.
Step 1: Express cot(x) in terms of sine and cosine.
cot(x) is the reciprocal of tan(x), and tan(x) can be written as sin(x) / cos(x). Therefore, cot(x) can be expressed as cos(x) / sin(x).
Step 2: Apply the quotient rule.
To differentiate cot(x) = cos(x) / sin(x), we’ll use the quotient rule, which states that if we have a function f(x) = u(x) / v(x), then its derivative f'(x) is given by:
f'(x) = (u'(x) * v(x) – u(x) * v'(x)) / v(x)^2
In this case, u(x) = cos(x) and v(x) = sin(x). Let’s differentiate these functions to find u'(x) and v'(x).
The derivative of cos(x) is -sin(x), and the derivative of sin(x) is cos(x).
Step 3: Apply the quotient rule formula.
Using the quotient rule formula, we can find the derivative of cot(x) = cos(x) / sin(x) as follows:
cot'(x) = ((-sin(x)) * sin(x) – cos(x) * cos(x)) / sin(x)^2
= (-sin^2(x) – cos^2(x)) / sin^2(x)
Step 4: Simplify the derivative expression.
Now, we can simplify the expression further. Recall the Pythagorean identity: sin^2(x) + cos^2(x) = 1. We can rearrange this to sin^2(x) = 1 – cos^2(x).
cot'(x) = (-(1 – cos^2(x)) – cos^2(x)) / sin^2(x)
= (-1 + 2cos^2(x)) / sin^2(x)
Therefore, the derivative of cot(x) with respect to x is (-1 + 2cos^2(x)) / sin^2(x).
It’s worth noting that this expression can also be written in terms of sin(x) instead of cos(x), using the identity 1 + cot^2(x) = csc^2(x).
More Answers:
A Guide to Understanding and Applying the Factor Theorem in AlgebraA Guide to Testing X-Axis Symmetry in Graph Equations
A Powerful Technique for Integrals | Understanding Integration by Parts in Calculus