How to Find the Derivative of cot^(-1)(x) Using the Chain Rule and Trigonometric Identities

derivative of cot^-1

To find the derivative of cot^(-1) (also known as arccot), we can use the chain rule and the derivative of the inverse trigonometric function

To find the derivative of cot^(-1) (also known as arccot), we can use the chain rule and the derivative of the inverse trigonometric function. Here’s the step-by-step process:

Let’s define a function, f(x) = cot^(-1)(x).
To find the derivative, we need to find f'(x).

Step 1: Recall the relationship between cot(x) and arccot(x):
cot(arccot(x)) = x

Step 2: Rewrite the function using the relationship above:
cot(f(x)) = x

Step 3: Differentiate both sides of the equation with respect to x:
– csc^2(f(x)) * f'(x) = 1

Step 4: Solve for f'(x):
f'(x) = 1 / csc^2(f(x))

Step 5: Convert the function back to cot^(-1)(x):
f'(x) = 1 / csc^2(cot^(-1)(x))

Step 6: Simplify using trigonometric identities:
Recall that csc(x) is the same as 1/sin(x).
So, we can rewrite the equation as:
f'(x) = 1 / (1/sin^2(cot^(-1)(x)))

Step 7: Apply the Pythagorean identity (sin^2(x) + cos^2(x) = 1):
f'(x) = sin^2(cot^(-1)(x))

Therefore, the derivative of cot^(-1)(x) is sin^2(cot^(-1)(x)).

It’s worth noting that this derivative can also be expressed using trigonometric identities. For example, using the identity sin^2(x) = 1 – cos^2(x):
f'(x) = 1 – cos^2(cot^(-1)(x))

Remember to always check your work and simplify further if necessary.

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