Understanding the Derivative of Arctan(x) with Respect to x | Explained Step by Step for Beginners

d/dx arctan(x)

To find the derivative of arctan(x) with respect to x, we can use the chain rule of differentiation

To find the derivative of arctan(x) with respect to x, we can use the chain rule of differentiation. The chain rule states that if we have a composition of functions, then the derivative of the composite function is the product of the derivative of the outer function and the derivative of the inner function.

Let’s start by differentiating the arctan function, which is the outer function. The derivative of arctan(x) with respect to x can be found using the fact that the derivative of the inverse function is equal to 1 divided by the derivative of the original function. In this case, the derivative of arctan(x) is 1 divided by the derivative of tan(x).

The derivative of tan(x) can be found using the quotient rule of differentiation. The quotient rule states that if we have u(x)/v(x), where u(x) and v(x) are differentiable functions, then the derivative can be computed using the formula (v(x) * u'(x) – u(x) * v'(x)) / v(x)^2.

In this case, u(x) = 1 and v(x) = tan(x). Thus, we have:
u'(x) = 0 (the derivative of a constant is 0)
v'(x) = sec^2(x) (the derivative of tan(x) is sec^2(x))

Using the quotient rule and substituting u'(x) and v'(x) into the formula, we get:
(tan(x) * 0 – 1 * sec^2(x)) / (tan(x))^2

Simplifying this expression, we have:
-sec^2(x) / tan^2(x)

Now, we can substitute this result back into the original derivative of arctan(x) formula. We have:
1 / (-sec^2(x) / tan^2(x))

To simplify this further, we multiply by the reciprocal of -sec^2(x) / tan^2(x). The reciprocal of -sec^2(x) / tan^2(x) is -tan^2(x) / sec^2(x). Therefore, we have:
1 / (-sec^2(x) / tan^2(x)) * (-tan^2(x) / sec^2(x))

Simplifying this expression,
= -tan^2(x) / sec^2(x) * (-sec^2(x) / tan^2(x))
= tan^2(x) / sec^2(x) * sec^2(x) / tan^2(x)
= 1

Hence, the derivative of arctan(x) with respect to x is equal to 1, which means that the slope of the tangent line to the curve y = arctan(x) at any given point is equal to 1.

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