Question

Differentiate the following funetions: \(y=\sqrt{a^{2}+x^{2}}\)

Short answer

Answer: The derivative of this function with respect to \(x\) is \(\frac{dy}{dx}=\frac{x}{\sqrt{a^{2}+x^{2}}}\).

Step-by-step solution

Rewrite the square root function

Rewrite the square root function as an exponent to make the differentiation easier. The square root of a function can be represented as raising the function to the power of \(\frac{1}{2}\). So we have: \(y=\left(a^{2}+x^{2}\right)^{\frac{1}{2}}\)

Apply the chain rule

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In this case, the outer function is \(f(u)=u^{\frac{1}{2}}\), and the inner function is \(g(x)=a^{2}+x^{2}\). Apply the chain rule: \(\frac{dy}{dx}=\frac{d}{dx}\left(\left(a^{2}+x^{2}\right)^{\frac{1}{2}}\right)=\frac{d}{du}\left(u^{\frac{1}{2}}\right)\frac{d}{dx}(a^{2}+x^{2})\)

Differentiate the outer and inner functions

We will now differentiate both the outer and inner functions separately. For the outer function: \(\frac{d}{du}\left(u^{\frac{1}{2}}\right)=\frac{1}{2}u^{-\frac{1}{2}}\) For the inner function: \(\frac{d}{dx}(a^{2}+x^{2})=2x\) Now substitute these derivatives back into the chain rule expression: \(\frac{dy}{dx}=\frac{1}{2}u^{-\frac{1}{2}}\cdot 2x\)

Substitute u back to the original function and simplify

Now, we'll substitute \(u\) back with the original function \(a^{2}+x^{2}\) and simplify the expression: \(\frac{dy}{dx}=\frac{1}{2}(a^{2}+x^{2})^{-\frac{1}{2}}\cdot 2x\) Simplify the expression: \(\frac{dy}{dx}=\frac{x}{\sqrt{a^{2}+x^{2}}}\)

Final Answer

The derivative of the given function with respect to \(x\) is: \(\frac{dy}{dx}=\frac{x}{\sqrt{a^{2}+x^{2}}}\)

Key concepts

Chain Rule

In calculus, the chain rule is an essential tool for differentiating composite functions. It allows us to find the derivative of a complex function by breaking it down into simpler parts. A composite function is a function made up of two or more other functions, where the output of one function becomes the input of another. The chain rule states: If we have a composite function \( y = f(g(x)) \), then its derivative \( \frac{dy}{dx} \) is equal to \( \frac{df}{dg} \times \frac{dg}{dx} \).

• Identify the outer function \(f\) and the inner function \(g\).
• Differentiate the outer function concerning the inner function \(\frac{df}{dg}\).
• Differentiate the inner function \(g\) concerning \(x\), which gives \(\frac{dg}{dx}\).
• Multiply these derivatives to get the final derivative \( \frac{dy}{dx} \).

In our original problem, we used the chain rule to differentiate \( y = \sqrt{a^2 + x^2} \), helping us handle its complex composition effectively.

Composite Function

Composite functions are formed when one function is applied inside another. They are denoted by \( f(g(x)) \), meaning that \( g(x) \) is an input to \( f(x) \). In the context of differentiation, identifying composite functions is crucial because they often require the use of the chain rule.

In the exercise, the function \( y = \sqrt{a^2 + x^2} \) is a classic example of a composite function. Here, \( f(u) = u^{1/2} \) is the outer function representing a square root, and \( g(x) = a^2 + x^2 \) is the inner function.

When tackling differentiation, spotting these functions allows us to simplify the process significantly by applying the chain rule. By breaking down into their respective outer and inner parts, we can manage even complex expressions with relative ease.

Derivative of Functions

The derivative of a function is a fundamental concept in calculus that measures how a function changes as its input changes. It provides the rate at which a function is increasing or decreasing at any given point. In simpler terms, it's about finding the slope of the function's graph at a particular point.

For our given exercise, differentiating \( y = \sqrt{a^2 + x^2} \) involves determining how quickly the value of \( y \) changes with a change in \( x \). We utilized the chain rule since the function is composite.

The steps to find a derivative generally involve identifying a suitable rule (like the power rule, product rule, quotient rule, or chain rule) and applying it systematically. In our case, after rewriting the expression \( \sqrt{a^2 + x^2} \) as \( (a^2 + x^2)^{1/2} \), we employed the chain rule to manage the differentiation neatly by separating the inner and the outer functions.

Calculus

Calculus is a branch of mathematics that studies change. It is broadly divided into two main parts: differentiation and integration. Differentiation deals with continuously changing quantities, determining rates of change, while integration focuses on accumulation and area under curves.

The application of calculus in solving the exercise helps us understand how a function behaves as the input changes. Differentiation, the main focus of our problem, allows us to analyze the function \( y = \sqrt{a^2 + x^2} \) to find how its values vary with respect to \(x\).

• Identifying derivatives for rates of change is critical in various fields like physics, engineering, and economics.
• The chain rule, as we've seen, is one powerful technique within differentiation, crucial for handling complex, nested functions.

Understanding calculus concepts like derivatives and the basics of the chain rule allows you to decipher more intricate math problems, preparing you for advanced mathematical challenges.