Problem 51 - Single Variable Calculus

Question

Find the derivative of the function. $$ f(x)=\tan ^{-1} x^{2} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = \tan^{-1}(x^2)\) is: $$ f'(x) = \frac{2x}{1 + x^4} $$

Step 1: Identify the inner and outer functions

The inner function is the squared term, and the outer function is the arctangent function. So we have: Inner function: $$ u(x) = x^2 $$ Outer function: $$ g(u) = \tan^{-1}(u) $$

Step 2: Differentiate the outer function with respect to u

Using the derivative of the arctangent and keeping in mind that we need to differentiate with respect to u, we get: $$ \frac{dg}{du} = \frac{1}{1 + u^2} $$

Step 3: Differentiate the inner function with respect to x

The inner function is just the squared term, so its derivative is: $$ \frac{du}{dx} = 2x $$

Step 4: Apply the chain rule

Now we apply the chain rule by multiplying the derivatives obtained in steps 2 and 3: $$ \frac{df}{dx} = \frac{dg}{du} \cdot \frac{du}{dx} = \frac{1}{1 + u^2} \cdot 2x $$

Step 5: Substitute the inner function back into the derivative

Finally, substitute the inner function, \(u(x) = x^2\), back into the expression for the derivative: $$ \frac{df}{dx} = \frac{2x}{1 + (x^2)^2} = \frac{2x}{1 + x^4} $$ The derivative of the function \(f(x) = \tan^{-1}(x^2)\) is: $$ f'(x) = \frac{2x}{1 + x^4} $$

Key Concepts

Chain Rule

The chain rule is an essential concept in calculus that helps us find the derivative of composite functions. A composite function is one where a function is applied to the result of another function. In simpler terms, if you have a function within a function, you're dealing with a composite function.
To apply the chain rule, you identify two parts of the composite function: the outer function and the inner function.

• The **outer function** operates on the result of the inner function.
• The **inner function** is the part of the composite that provides the input for the outer function.

The chain rule then states that the derivative of the composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In mathematical notation, this is expressed as:\[ (f(g(x)))' = f'(g(x)) \cdot g'(x) \] Let's consider our exercise with \( f(x) = \tan^{-1}(x^2) \):

• Inner function: \(u(x) = x^2\)
• Outer function: \(g(u) = \tan^{-1}(u)\)

The chain rule helps us break this problem down into more manageable parts by differentiating each separately before combining them. This method not only simplifies the differentiation of complex functions but also reinforces understanding by methodically stepping through each part.

Arctangent Function

The arctangent function, denoted as \(\tan^{-1}(x)\) or \(\text{arctan}(x)\), is the inverse of the tangent function. It plays a crucial role in trigonometry and calculus. The primary range for the arctangent function is typically between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
When dealing with the differentiation of the arctangent function, it is important to understand its derivative. The derivative of the arctangent function with respect to its argument is:\[\frac{d}{dx}[\tan^{-1}(x)] = \frac{1}{1 + x^2}\]This expression signifies the rate of change of the arctangent concerning its input value, providing a useful formula in calculus problems.
In our exercise, the arctangent operates on \(x^2\), which means the derivative must adapt to the inner function \(x^2\). Thus, when finding the derivative involving the arctangent, this fundamental derivative rule forms a backbone, just as seen in our step-by-step process.

Functions Differentiation

Differentiation is the mathematics tool that allows us to determine the rate at which a function changes at any given point. It's a fundamental element of calculus and finds applications in numerous fields. When differentiating functions, breaking down the problems into simpler derivative calculations can be extremely helpful.
In the case of composite functions, as we have with \( f(x) = \tan^{-1}(x^2) \), differentiation requires strategy and methodical steps:

• **Identify the structure**: Recognize inner and outer functions, as this helps apply rules like the chain rule effectively.
• **Calculate the derivatives separately**: Compute the derivative of each function independently before merging the results, as seen with differentiating \(x^2\) and \(\tan^{-1}(u)\).
• **Combine results appropriately**: Use multiplication of derivatives, interaction terms, and substitution where necessary, like substituting \(u = x^2\) back into our expression.

This structured approach ensures clarity and helps avoid common pitfalls in differentiation such as misapplication of rules or overlooking components of complex functions. With practice, these steps become intuitive, allowing for efficient and accurate derivatives.