Understanding the Derivative of Trigonometric Functions

The question asks for the derivative of the expression \(\sec^2(\tan^{-1}x)\) with respect to \(x\). To solve this, we can use a trigonometric identity to simplify the expression before differentiation.

Simplifying the Expression \(\sec^2(\tan^{-1}x)\)

We know the trigonometric identity:

\(\sec^2 \theta = 1 + \tan^2 \theta\)

Let \(\theta = \tan^{-1}x\). This means that \(\tan \theta = x\).

Now, substitute \(\theta = \tan^{-1}x\) into the identity:

\(\sec^2(\tan^{-1}x) = 1 + \tan^2(\tan^{-1}x)\)

Since \(\tan(\tan^{-1}x) = x\) for all real numbers \(x\), we can simplify the right side:

\(\sec^2(\tan^{-1}x) = 1 + (x)^2\)

So, the expression simplifies to \(1 + x^2\).

Calculating the Derivative

Now we need to find the derivative of \(1 + x^2\) with respect to \(x\).

Let \(y = 1 + x^2\). We want to find \(\frac{dy}{dx}\).

We can find the derivative term by term using the sum rule and the power rule for differentiation:

\(\frac{d}{dx}(1 + x^2) = \frac{d}{dx}(1) + \frac{d}{dx}(x^2)\)

The derivative of a constant (like 1) with respect to \(x\) is 0. So, \(\frac{d}{dx}(1) = 0\).

The derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\) (Power Rule). For \(x^2\), \(n=2\). So, \(\frac{d}{dx}(x^2) = 2x^{2-1} = 2x^1 = 2x\).

Combining these results:

\(\frac{dy}{dx} = 0 + 2x = 2x\)

Thus, the derivative of \(\sec^2(\tan^{-1}x)\) with respect to \(x\) is \(2x\).

Step-by-Step Differentiation Summary

Recognize the expression \(\sec^2(\tan^{-1}x)\).

Use the identity \(\sec^2 \theta = 1 + \tan^2 \theta\) with \(\theta = \tan^{-1}x\).

Simplify \(\tan(\tan^{-1}x)\) to \(x\).

Rewrite the original expression as \(1 + x^2\).

Differentiate \(1 + x^2\) with respect to \(x\).

The derivative of \(1\) is \(0\).

The derivative of \(x^2\) is \(2x\).

The total derivative is \(0 + 2x = 2x\).

Revision Table: Key Derivatives

Additional Information: Inverse Trigonometric Functions

Inverse trigonometric functions, like \(\tan^{-1}x\) (also written as arctan \(x\)), are the inverse functions of the trigonometric functions. They are used to find the angle when the value of the trigonometric ratio is known.

For \(\tan^{-1}x\), the domain is all real numbers (\(\mathbb{R}\)), and the principal range is \((-\frac{\pi}{2}, \frac{\pi}{2})\). The property \(\tan(\tan^{-1}x) = x\) holds for all \(x\) in the domain of \(\tan^{-1}x\), which is \(\mathbb{R}\).

Understanding the relationships between trigonometric functions and their inverses, as well as basic identities, can greatly simplify complex differentiation problems.