Question

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. \(f(x)=\frac{x^{2}+c^{2}}{x^{2}-c^{2}}, \quad c\) is a constant

Short answer

The derivative of the given function is \(f'(x)=\frac{-4c^{2}x}{(x^{2}-c^{2})^{2}}\).

Step-by-step solution

Identify \(u\) and \(v\)

In this exercise, identify \(u=x^{2}+c^{2}\), and, \(v=x^{2}-c^{2}\).

Find \(u'\) and \(v'\)

Find the derivative of \(u\) and \(v\). The derivatives of \(u\) and \(v\) are \(u'=2x\) and \(v'=2x\) respectively.

Apply the quotient rule

Now, apply the quotient rule and substitute \(u, u', v, v'\). Using the formula given by the quotient rule \(f'(x) = \frac{u'v - uv'}{v^{2}}\) we get \(f'(x)=\frac{(2x(x^{2}-c^{2})) - ((x^{2}+c^{2})2x)}{(x^{2}-c^{2})^{2}}\).

Simplify the expression

The expression simplifies to \(f'(x)=\frac{(-4c^{2}x)}{(x^{2}-c^{2})^{2}}\).

Key concepts

Quotient Rule

Understanding how to find the derivative of a ratio of two functions is crucial in calculus. This is where the quotient rule comes into play. It is a formal rule for differentiating problems where one function is divided by another. The quotient rule states that the derivative of a function represented as the quotient \( \frac{u}{v} \) is given by \( \frac{u'v - uv'}{v^2} \), where \( u \) and \( v \) are differentiable functions of \( x \) and \( u' \) and \( v' \) are their respective derivatives.

When applying the quotient rule, it's essential to differentiate the numerator and the denominator separately before combining them according to the rule. It's an invaluable tool when dealing with complex rational functions, which are frequent in various mathematical and real-world applications.

Derivative of Algebraic Functions

An algebraic function includes operations like addition, subtraction, multiplication, division, and taking roots of variables and constants. When tasked with finding the derivative of algebraic functions such as \( \frac{x^2 + c^2}{x^2 - c^2} \) where \( c \) is a constant, you are actually dealing with the application of several differentiation rules. To differentiate efficiently, remember that constants come out of derivatives unchanged (such as \( c^2 \) in our function) and the power rule (\( (x^n)' = nx^{n-1} \)) is often used for polynomial terms. Also, the distinction between constants and variables is essential—constants have a derivative of zero.

When a constant is part of your algebraic function, like in our exercise, it remains constant through the differentiation process. By applying the differentiation rules, including the power rule, to the variable parts and considering the constant, you build a derivative function that describes the rate at which the original function changes with respect to \( x \).

Differentiation

Differentiation is a fundamental operation in calculus that measures how a function changes as its input changes. Essentially, it determines the rate at which a quantity changes. The derivative of a function at a point is the slope of the tangent line at that point on the graph of the function. For functions like \( f(x) = \frac{x^2 + c^2}{x^2 - c^2} \) discussed earlier, differentiation allows us to compute the rate of change efficiently.

Practically speaking, to differentiate a function means to apply rules such as the power rule, product rule, quotient rule, and chain rule, depending on the function's form. The process includes simplifying the function if possible and then systematically taking derivatives of its parts according to these rules. The essence of differentiation is to provide a precise mathematical means of computing and predicting changes, which is invaluable in fields such as physics, engineering, economics, and beyond.