Derivative of Inverse Tangent Function (tan^-1(x)) | Formula, Derivation, and Result

tan^-1(x) derivative

To find the derivative of the inverse tangent function (tan^-1(x)), we can use the formula for the derivative of an inverse function:

If y = tan^-1(x), then x = tan(y)

To find the derivative of the inverse tangent function (tan^-1(x)), we can use the formula for the derivative of an inverse function:

If y = tan^-1(x), then x = tan(y).

Now, we can differentiate both sides with respect to y using implicit differentiation:

d/dy(x) = d/dy(tan(y))

1 = sec^2(y) * dy/dy

Since sec^2(y) is equal to 1 + tan^2(y), we can rewrite the equation as:

1 = (1 + tan^2(y)) * dy/dy

Simplifying, we get:

1 = 1 + tan^2(y) * dy/dy

Subtracting 1 from both sides, we have:

tan^2(y) * dy/dy = 0

Dividing both sides by tan^2(y), we get:

dy/dy = 0 / tan^2(y)

Since dy/dy is the derivative of y with respect to y, it is simply 1.

So, we have:

1 = 0 / tan^2(y)

Multiplying both sides by tan^2(y), we get:

tan^2(y) = 0

Now, we can solve for y. Taking the square root of both sides, we have:

tan(y) = 0

Since the tangent of an angle only equals zero at certain angles (e.g., 0 degrees, 180 degrees), we have:

y = 0 degrees + n*180 degrees

Now, let’s go back to the original equation x = tan(y). Replacing y with 0 degrees + n*180 degrees, we have:

x = tan(0 degrees + n*180 degrees)

Since the tangent function has a period of 180 degrees, we can simplify this to:

x = tan(0 degrees)

The tangent of 0 degrees is equal to 0. Therefore, the solution is:

x = 0

Therefore, the derivative of tan^-1(x) is 0 (or 0 if we consider the input in radians).

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