Question

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the indicated point. $$ y^{2}=\frac{x^{2}-9}{x^{2}+9} $$

Short answer

The derivative of the expression, obtained through implicit differentiation, is \( dy/dx = \frac{18x}{y(x^2 + 9)^2} \)

Step-by-step solution

Differentiate implicitly

We start by differentiating each side of the equation \( y^{2}=\frac{x^{2}-9}{x^{2}+9} \) with respect to \( x \). Applying the chain rule and the quotient rule, we obtain \(2y \frac{dy}{dx} = \frac{(2x)(x^2 + 9) - (2x)(x^2 - 9)}{(x^2 + 9)^2} \). Simplifying the right-hand side gives \(2y \frac{dy}{dx} = \frac{36x}{(x^2 + 9)^2} \)

Solve for \(dy/dx\)

Next step is to solve for \( dy/dx \). To do this you should isolate \( dy/dx \) on one side of the equation. You can accomplish this by dividing both sides of the equation by \( 2y \), so we have there: \( dy/dx = \frac{36x}{2y(x^2 + 9)^2} \). Simplifying this expression results in \( dy/dx = \frac{18x}{y(x^2 + 9)^2} \)

Evaluate the derivative at the given point

The exercise does not specify a point at which to evaluate the derivative, so no calculation is possible in this step.

Key concepts

Chain Rule

Understanding the chain rule is crucial for performing complex forms of differentiation, such as implicit differentiation. Put simply, the chain rule is a formula to compute the derivative of a composite function. When two or more functions are combined, like nesting dolls, the chain rule allows us to differentiate the entire combination by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

In the context of our exercise, where we differentiate the function y², we treat y as an implicit function of x (i.e., y(y(x))). Applying the chain rule, the derivative of y² with respect to x is 2y(dy/dx), where dy/dx represents the derivative of y with respect to x.

This step is critical because it sets the foundation for finding the derivative of the entire expression on the right side of the equation.

Quotient Rule

The quotient rule is a technique used when differentiating ratios of functions. It is particularly helpful when a function is divided by another, as with ratios or fractions. The standard form of the quotient rule is given as: if you have a function u/v, its derivative u'/v will be (vu' - uv') / v², where u' and v' are derivatives of u and v respectively with respect to x.

In the given exercise, the right side of our initial equation is such a ratio between the functions x² - 9 and x² + 9. To differentiate it, we apply the quotient rule which simplifies the expression considerably. We first find the derivatives of the numerator and the denominator, then subtract the product of the denominator and the derivative of the numerator from the product of the numerator and the derivative of the denominator. Lastly, we divide the result by the square of the denominator to get our right side derivative as a function of x.

Derivative Evaluation

Once the derivative has been found, it can often be important to evaluate it at a specific point to find the rate of change at that location. This is known as derivative evaluation. It involves substituting a particular value of x into the derivative formula to find the slope of the tangent line to the curve at that point.

However, in our case, the exercise did not provide a specific point to evaluate the derivative. Typically, if a point (x₀, y₀) is provided, x₀ would be substituted into the derivative, and if the expression for the derivative includes y, y₀ must also be known or solvable from the original equation. This final evaluation step is where the abstract process of differentiation yields concrete, practical information about the behavior of the function at a particular point.