Derivative of Cot Inverse

The derivative of cot inverse, also known as the derivative of arccot, gives the value of the differentiation of cot inverse x which is the rate change of the change in the function cot inverse with respect to the variable. The derivative of cot inverse x is equal to -1/(1 + x²). We can evaluate the differentiation of cot inverse x using different methods of derivatives such as the first principle of derivatives, that is, the definition of limits and the implicit differentiation method. Hence, we can say that the rate of change of cot inverse is given by -1/(1 + x²).

Further, in this article, we will explore the derivative of cot inverse and derive its formula using various differentiation methods. We will also solve various examples related to the concept of differentiation of cot inverse for a better understanding.

The derivative of cot inverse is equal to -1/(1 + x²) which is mathematically written as d(cot^-1)/dx = -1/(1 + x²) = d(arccot)/dx. We can evaluate the cot inverse derivative using various differentiation methods including the first principle of differentiation and implicit differentiation method. As we know that differentiation of a function is a process of finding the rate of change of a function with respect to the change in the variable. We know that the derivative of tan inverse x is equal to 1/(1 + x²), therefore the derivative of cot inverse is the negative of the derivative of tan inverse. Let us go through the formula of the derivative of cot inverse x in the next section.

The formula for the differentiation of cot inverse is given by, d(cot^-1x)/dx = -1/(1 + x²). We can easily memorize the formula of the derivative of arccot using the fact that it is negative of the derivative of arctan. Hence, we can write it as d(cot^-1x)/dx = - d(tan^-1x)/dx = -1/(1 + x²). The image below shows the formula for the derivative of cot inverse x.

Now that we know that the derivative of cot inverse x is equal to d(cot^-1x)/dx = -1/(1 + x²), we will prove it using the method of implicit differentiation. Assume y = cot^-1x, then taking cot on both sides of the equation, we have cot y = x. Now, differentiating both sides of the equation cot y = x with respect to x, we have

cot y = x

⇒ -cosec²y × dy/dx = dx/dx

⇒ dy/dx = 1 × -1/cosec²y

⇒ dy/dx = -1/cosec²y

Now, using the trigonometric identity 1 + cot²A = cosec²A, we have

dy/dx = -1/cosec²y

⇒ dy/dx = -1/(1 + cot²y)

= -1/(1 + x²) --- [Because cot y = x]

⇒ d(cot^-1x)/dx = -1/(1 + x²)

Therefore, we have proved that the derivative of cot inverse x is equal to -1/(1 + x²).

In this section, we will derive the formula for the derivative of cot inverse using the first principle of derivatives. We will use certain differentiation formulas and inverse trigonometric identities to prove the result such as:

• cot^-1x + tan^-1x = π/2 ⇒ cot^-1x = π/2 - tan^-1x
• f'(x) = lim_h→0 [f(x+h) - f(x)]/h
• d(tan^-1)/dx = 1/(1 + x²)

Using the above formulas, we have

d(cot^-1x)/dx = lim_h→0 [cot^-1(x+h) - cot^-1x]/h

= lim_h→0 [π/2 - tan^-1(x+h) - (π/2 - tan^-1x)]/h

= lim_h→0 [π/2 - tan^-1(x+h) - π/2 + tan^-1x)]/h

= lim_h→0 [- tan^-1(x+h) + tan^-1x)]/h

= - lim_h→0 [tan^-1(x+h) - tan^-1x)]/h

= - d(tan^-1x)/dx ---- [Using the definition of derivatives using limits]

= - 1/(1 + x²) --- [Because the derivative of arctan is 1/(1 + x²)]

Hence, the derivative of cot inverse x is equal to - 1/(1 + x²) using the first principle of derivatives.

Important Notes on Derivative of Cot Inverse

• The derivative of cot inverse x is equal to d(cot^-1x)/dx = -1/(1 + x²).
• We can derive the cot inverse derivative using the implicit differentiation and first principle of derivatives.

☛ Related Topics:

• Derivative of Cos Inverse
• Derivative of Sin Inverse
• Derivative of Arcsec

What is the Derivative of Cot Inverse in Calculus?

The derivative of cot inverse, also known as the derivative of arccot, gives the value of the differentiation of cot inverse x which is the rate change of the change in the function cot inverse with respect to the variable. The derivative of cot inverse is equal to -1/(1 + x²).

What is the Formula for the Derivative of Cot Inverse x?

The formula for the differentiation of cot inverse is given by, d(cot^-1x)/dx = -1/(1 + x²). We can also write it as d(cot^-1x)/dx = - d(tan^-1x)/dx = -1/(1 + x²)

How to Find the Differentiation of Cot Inverse?

We can evaluate the differentiation of cot inverse using the first principle of derivatives and the method of implicit differentiation.

How is Derivative of Cot Inverse Related to the Derivative of Tan Inverse?

The derivative of cot inverse is the negative of the derivative of tan inverse. Therefore, we can write it as d(arccot)/dx = -d(arctan)/dx = -1/(1 + x²).

What is the Derivative of Tan Inverse x with Respect to Cot Inverse x?

On evaluation, we have d(tan^-1x)/d(cot^-1x) = -1, that is, the derivative of tan inverse x with respect to cot inverse x is equal to -1.

What is the Derivative of Cot Inverse x Square?

The derivative of cot inverse x square is given by d(cot^-1(x²))/dx = -2x / (1 + x⁴). We can calculate this derivative using the derivative of cot inverse x and chain rule method of differentiation.