Question

Find \(d y / d x\). (Treat \(a\) and \(r\) as constants.) $$y^{3 / 2}+x^{3 / 2}=16$$

Short answer

\(\frac{dy}{dx} = -\frac{x^{1/2}}{y^{1/2}}\)

Step-by-step solution

Differentiate both sides with respect to x

Apply the derivative with respect to x to both sides of the equation using the chain rule for the term with y, since y is a function of x, and the power rule for the term with x.

Differentiate the left-hand side

Differentiate each term individually. For the first term, use chain rule \( \frac{d}{dx}(y^{3/2}) = \frac{3}{2}y^{1/2}\frac{dy}{dx} \). For the second term, use power rule \( \frac{d}{dx}(x^{3/2}) = \frac{3}{2}x^{1/2} \)..

Differentiate the right-hand side

Since the right-hand side of the equation is a constant, its derivative with respect to x is 0.

Isolate \(\frac{dy}{dx}\)

After differentiating both sides, isolate \(\frac{dy}{dx}\) on one side of the equation to solve for it.

Solve for \(\frac{dy}{dx}\)

Manipulate the equation from the previous step to express \(\frac{dy}{dx}\) in terms of x and y.

Key concepts

Chain Rule

When dealing with functions within functions—such as a variable 'y' that itself depends on another variable 'x'—implicit differentiation is a valuable technique that requires the chain rule for execution. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Essentially, it's the mathematical way of saying 'first, take the derivative of the outside layer, and then multiply it by the derivative of the inside layer.'

For example, for a function like \(y^3\), where \(y\) itself is a function of \(x\), we first take the derivative of the outer function, which gives us \(3y^2\), and then multiply that by the derivative of \(\frac{dy}{dx}\), resulting in \(3y^2 \frac{dy}{dx}\). This step is pivotal because it aligns with the hypothesis that 'y' changes as 'x' changes—hence, we cannot treat 'y' as a constant.

Power Rule

The power rule is one of the most straightforward rules in differentiation, suitable for dealing with functions where the variable is raised to a power. It states that the derivative of \(x^n\) is \(nx^{n-1}\). In practice, simply multiply the power by the coefficient and subtract one from the power to find the new exponent.

Applying this rule to an expression like \(x^{3/2}\), we get \(\frac{3}{2}x^{3/2 - 1} = \frac{3}{2}x^{1/2}\), demonstrating the rule's effectiveness in quickly finding derivatives for power terms. Remember, however, that this only applies when 'x' is the variable, not when dealing with a function of another variable like 'y'.

Derivative of a Constant

Understanding the concept that constants do not change as the variable changes is crucial in differentiation. Thus, the derivative of a constant is zero. Whether the constant is standalone or the result of a complicated calculation, if it does not involve the variable we are differentiating with respect to (in this case, \(x\)), its derivative is zero.

This property allows us to simplify the process of differentiating equations by instantly setting the derivative of any constant term to zero without further calculation. For instance, in the equation \(y^3 + x^3 = 16\), if we differentiate both sides with respect to \(x\), the term \(16\), being a constant, would have a derivative of zero.

Solving Derivatives

To solve for a derivative means to isolate the derivative (let's say \(\frac{dy}{dx}\)) on one side of the equality to express it explicitly in terms of other variables and constants in the equation. This is often required in implicit differentiation where the derivative of \(y\) with respect to \(x\) needs to be determined.

In an equation where both \(y\) and \(x\) are raised to powers, applying differentiation rules results in expressions that include \(\frac{dy}{dx}\). To solve for \(\frac{dy}{dx}\), one would typically gather all the terms involving \(\frac{dy}{dx}\) on one side and factor it out. This procedure allows us to find the rate of change of one variable with respect to another, providing insights into their relationship within a given context.