Examples on Derivative of Arctan x

Example 1: Find the derivative of the function f(x) = arctan(3x).

Solution:

We will use the chain rule, which states that if g(x) is differentiable at x and f(x) = arctan (g(x)), then the derivative f'(x) is given by:

f'(x) = g'(X)/(1+[g(x)]²)

In this case, g(x) = 3x, so g'(X) = 3. Applying the chain rule formula:

f'(x) = 3/(1+(3x)²)

f'(x) = 3/(1+9x²)

Example 2: Find the derivative of the function h(x) = tan^-1(x/2)

Solution:

We will use the chain rule, according which f(x) = tan^-1(g(x)), then the derivative f'(x) is given by:

f'(x) = g'(X)/(1+[g(x)]²)

In this case, g(x) = x/2, so g'(X) = 1/2. Applying the chain rule formula:

f'(x) = (1/2)/(1+(x/2)²)

f'(x) = (1/2)/(1+x²/4)

Simplifying we get,

f'(x) = 2/(4+x²)

Example 3: Find the derivative of f(x) = arctan (2x²)

Solution:

We will use the chain rule, which states that if g(x) is differentiable at x and f(x) = arctan (g(x)), then the derivative f'(x) is given by:

f'(x) = g'(X)/(1+[g(x)]²)

In this case, g(x) = 2x², so g'(X) = 4x.

Applying the chain rule formula:

f'(x) = 4x/(1+(2x²)²)

f'(x) = 4x/(1+4x⁴)

f'(x) = d/dx(arctan (2x²)) = 4x/(1+4x⁴)

Derivative of the arc tangent function is denoted as tan^-1(x) or arctan(x). It is equal to  1/(1+x²). Derivative of arc tangent function is found by determining the rate of change of arc tan function with respect to the independent variable. The technique for finding derivatives of trigonometric functions is referred to as trigonometric differentiation.

In this article, we will learn about the derivative of arc tan x and its formula including the proof of the formula. Other than that, we have also provided some solved examples for better understanding.