Calculus often appears complex, but breaking down derivatives is a vital step in mastering it. Think of cot x as a ratio in a right triangle, the side next to x divided by the side opposite x, or cos of x divided by sin of x. In this blog, we’re going to find the derivative of cot x. By the time you reach the end, you’ll have a clear grasp of this essential concept in mathematics.
Finding the Derivative
When we want to find the derivative of cot x, we’re basically figuring out how it changes as x changes. The result is -1 times the square of cosec x, which might sound complex, but it’s just a way of measuring how the function varies with x.
Recommended Reading: What is the Derivative of sec x?
The formula for differentiation of cot x is,
d/dx (cot x) = -cosec2x (or)
(cot x)’ = -cosec2x
Let’s prove it step-by-step:
Step 1: Express cot x in terms of sine and cos:
cot x = cos x / sin x.
Step 2: Identify u and v for the quotient rule:
Let u = cos x.
Let v = sin x.
Step 3: Find the derivatives of u and v
a. Find du/dx:
Derivative of cos x is -sin x.
b. Find dv/dx:
Derivative of sin x is cos x.
Step 4: Apply the quotient rule
We know that cot x = (cos x)/(sin x).
So we assume that cot x = y
y = (cos x)/(sin x).
Step 5: Then by quotient rule derivative of y = y’,
y’ = [ sin x · d/dx (cos x) – cos x · d/dx (sin x)] / (sin^2 x)
= [sin x · (- sin x) – cos x (cos x)] / (sin^2 x)
= [-sin^2 x – cos^2 x] / (sin^2 x)
= -[-sin^2 x + cos^2 x] / (sin^2 x)
By one of the Pythagorean identities, cos^2 x + sin^2 x = 1. So
y’ = -1 / (sin^2 x) = -cosec^2 x
Recommended Reading: Understanding the Derivative of 2/x
Conclusion
Tackling calculus can feel like navigating through a maze, but understanding derivatives is like finding the key to unlock it. We’ve taken a deep dive into the derivative of cot x, picturing it as a ratio in a right triangle—cos of x divided by sin of x—which sheds light on its behavior as x changes. By following along step by step, you’ve unlocked a fundamental puzzle piece in the world of math.
