How To Find The Derivative Of F(X) = Tan^-1(X) Using Implicit Differentiation – Step-By-Step Explanation With Optimization Techniques.

d/dx[tan-1(x)]=

1/x^2+1

We can use implicit differentiation to find the derivative of the function f(x) = tan^-1(x):

Let y = tan^-1(x)

Taking the tangent of both sides, we get:

tan(y) = x

Now we can differentiate both sides with respect to x using the chain rule:

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

Multiplying both sides by cos^2(y):

cos^2(y) * sec^2(y) * dy/dx = cos^2(y)

Using the identity 1 + tan^2(y) = sec^2(y), we can substitute in:

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

Simplifying:

dy/dx = cos^2(y) / (1 + tan^2(y))

Substituting back in for y:

dy/dx = cos^2(tan^-1(x)) / (1 + tan^2(tan^-1(x)))

Using the Pythagorean identity sin^2 + cos^2 = 1, we can find that:

sin^2(tan^-1(x)) + cos^2(tan^-1(x)) = 1

Therefore:

cos^2(tan^-1(x)) = 1 – sin^2(tan^-1(x))

Substituting this back in:

dy/dx = (1 – sin^2(tan^-1(x))) / (1 + tan^2(tan^-1(x)))

Recalling the definition of tangent as the opposite over the adjacent, we can draw a right triangle with opposite side x and adjacent side 1, so that tan(tan^-1(x)) = x/1 = x. Similarly, we can draw a right triangle with opposite side 1 and adjacent side x to find that sin(tan^-1(x)) = 1 / sqrt(1 + x^2).

Substituting these back in:

dy/dx = (1 – (1 / (1 + x^2))) / (1 + x^2)

Simplifying, we get:

dy/dx = 1 / (1 + x^2)

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