Problem 15 - Calculus of a Single Variable

Question

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=2 \sqrt[4]{9-x^{2}} $$

Step-by-Step Solution

Answer

The derivative of the given function is \(y' = -x(9 - x^{2})^{-3/4}\)

Step 1: Convert the radical to exponential form

The given function is written as \(y = 2 \sqrt[4]{9-x^{2}}\). We can convert the 4th root into fractional exponent form. This becomes: \(y = 2(9-x^{2})^{1/4}\)

Step 2: Apply the power rule

Applying the power rule for derivatives (\(d/dx[a * f(x)^n] = n * a * f(x)^{n-1} f'(x)\)), we have \(y' = 1/4 * 2 (9 - x^{2})^{1/4 - 1} * -(2x)\). Note that \(f'(x)\) is the derivative of the function inside the parentheses, hence the \(-2x\)

Step 3: Simplify the derivative

Simplifying the derivative and replacing \(f'(x)\) with \(y'\) yields: \(y' = -x(9 - x^{2})^{-3/4}\)

Key Concepts

Power Rule for Derivatives

The power rule is a basic technique used to find the derivative of powers of x. It states that if you have a function of the form \( f(x) = x^n \) where \( n \) is any real number, then its derivative, labelled as \( f'(x) \) or \( \frac{d}{dx}f(x) \) is given by \( f'(x) = nx^{n-1} \). This rule significantly simplifies the process of differentiating polynomials.

For instance, if our function is \( y = x^4 \) then applying the power rule will give us the derivative \( y' = 4x^3 \) by multiplying the exponent by the coefficient and subtracting one from the exponent. But, what if the exponent isn't a whole number? The power rule still applies! This is exactly what we encounter when we differentiate functions with fractional or negative exponents. To successfully apply the power rule in these cases, we must be comfortable with rewriting expressions which brings us to the concept of radical to exponential form.

Radical to Exponential Form

To effectively work with derivatives of radical functions, it's often essential to rewrite them using fractional exponents because this puts the function into a form where the power rule can be applied directly.

Radical expressions like \( \sqrt[n]{x} \) can be converted to \( x^{1/n} \) easily, where \( n \) is the index of the root. This conversion allows for the direct application of the power rule. For example, a fourth root like \( \sqrt[4]{x} \) would become \( x^{1/4} \) in exponential form. If there are additional terms under the radical, such as \( \sqrt[4]{9-x^2} \) in the cited exercise, the entire expression under the root is raised to the fractional exponent: \( (9-x^2)^{1/4} \). Understanding this concept is pivotal, as proper reformulation of the function sets the stage for smooth differentiation.

Fractional Exponents

Fractional exponents are another way of expressing radicals, where the numerator of the fraction dictates the power to which the base is raised, and the denominator represents the type of root. In basic terms, \( x^{m/n} \) is the nth-root of \( x^m \) or vice-versa (\( (x^{1/n})^m \) or \( (x^m)^{1/n} \)).

This notation is not just for show; it simplifies many algebraic processes, including differentiation. For instance, when we encounter negative fractional exponents such as \( x^{-3/4} \) while differentiating, we must realize that this represents \( \frac{1}{x^{3/4}} \) or \( \frac{1}{\sqrt[4]{x^3}} \). When we differentiate an expression with a fractional exponent, we apply the power rule as usual, but we manage the fractional exponents carefully during simplification to end up with a neat result. This relates to our example where after applying the derivative power rule, \( (9 - x^{2})^{1/4} \) becomes \( (9 - x^{2})^{-3/4} \) upon differentiation.