Question

Evaluate the derivative of the following functions. $$f(w)=w^{2}-\tan ^{-1} w^{2}$$

Short answer

Question: Find the derivative of the function \(f(w) = w^2 - \tan^{-}_{1}(w^2)\) with respect to 'w'. Answer: The derivative of the given function is $$f'(w) = 2w - \frac{2w}{1+w^4}$$

Step-by-step solution

Identify the parts of the function

The function f(w) has two parts: 1. The power function: \(w^2\) 2. The inverse tangent function: \(-\tan^{-1}(w^2)\) We will differentiate each part separately.

Differentiate the first part of the function

To find the derivative of the first part, \(w^2\), we will apply the power rule: \(\frac{d}{dw}(w^2) = 2w\)

Differentiate the second part of the function

To find the derivative of the second part, \(-\tan^{-1}(w^2)\), we need to apply the chain rule: \(\frac{d}{dw}(-\tan^{-1}(w^2)) = -\frac{1}{1+(w^2)^2} \times \frac{d}{dw}(w^2)\) Using the power rule again for the inner function \(w^2\): \(\frac{d}{dw}(w^2) = 2w\) So, \(\frac{d}{dw}(-\tan^{-1}(w^2)) = -\frac{1}{1+(w^2)^2} \times (2w) = -\frac{2w}{1+w^4}\)

Combine the derivatives

Now we combine the derivatives of the two parts to find the derivative of the whole function: \(\frac{d}{dw}f(w) = 2w - \frac{2w}{1+w^4}\)

Final Answer

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

Key concepts

Power Rule

The power rule is one of the foundational rules for finding derivatives and is incredibly simple to apply. It helps us find the derivative of any function of the form \(x^n\), where \(n\) is any real number. The power rule states:

• \(\frac{d}{dx}(x^n) = nx^{n-1}\)

This means that you multiply the exponent \(n\) by the variable \(x\) raised to the power of \(n-1\).
For example, if you have the function \(w^2\), using the power rule, we find \(\frac{d}{dw}(w^2) = 2w^{2-1} = 2w\). This derivative tells us how \(w^2\) changes with respect to \(w\).
By applying the power rule, you can easily handle and differentiate polynomial functions.

Inverse Tangent Function

The inverse tangent function, commonly denoted as \(\tan^{-1} x\) or sometimes \(\arctan x\), is the inverse of the tangent function. It is used primarily in trigonometry and calculus to find an angle whose tangent is a given number. The derivative of the inverse tangent function is well-defined, and it follows a specific rule:

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

When differentiating the inverse tangent function as part of another function, such as in this exercise where we have \(-\tan^{-1}(w^2)\), you must also consider the chain rule because of the \(w^2\) inside the function. Understanding the derivative of the inverse tangent is crucial in solving problems involving angles and curves since it frequently appears in calculus problems that deal with real-world applications.

Chain Rule

The chain rule is a core concept in calculus, used to differentiate composite functions. When you have a function inside another function, the chain rule is the tool you need. The chain rule states:

• If \(y = g(f(x))\), then \(\frac{dy}{dx} = g'(f(x)) \cdot f'(x)\).

This rule helps us systematically find derivatives of functions that are nested within each other.
In the exercise, the function \(-\tan^{-1}(w^2)\) requires the chain rule because of the \(w^2\) inside the \(\tan^{-1}\) function. First, you take the derivative of the outer function \(\tan^{-1}\), which gives \(\frac{-1}{1+(w^2)^2}\). Then, multiply by the derivative of the inner function \(w^2\), which is \(2w\).
This results in \(-\frac{2w}{1+w^4}\). Learning the chain rule is essential for correctly finding the derivatives of nested functions in calculus.