Question

Use implicit differentiation to find\(\frac{d y}{d x}.\) $$x^{3}=\frac{x+y}{x

Short answer

Answer: The derivative is \(\frac{d y}{d x} = 4x^3 - 1\).

Step-by-step solution

Rewrite the equation

Instead of having a fraction, let's clear the fraction by multiplying both sides by \(x\). So, the new expression will be: $$x^4 = x + y$$

Apply implicit differentiation

Now we'll apply implicit differentiation with respect to \(x\) on both sides of the equation. $$\frac{d}{dx}(x^4) = \frac{d}{dx}(x + y)$$

Differentiate each term

To differentiate each term, remember that \(\frac{dy}{dx}\) represents the derivative of \(y\) with respect to \(x\). Notice that the left side only has terms in \(x\), while the right side has both \(x\) and \(y\) terms. $$4x^3 = 1 + \frac{dy}{dx}$$

Solve for \(\frac{d y}{d x}\)

Finally, we'll solve for \(\frac{d y}{d x}\). We simply need to isolate the term: $$frac{dy}{dx} = 4x^3 - 1$$ So the derivative is: $$\frac{d y}{d x} = 4x^3 - 1$$

Key concepts

Derivative of y with Respect to x

When working with functions that aren't explicitly solved for one variable in terms of another, such as in the equation presented in our exercise (\(x^4 = x + y\)), we turn to a powerful technique called implicit differentiation. This method allows us to find the derivative of one variable with respect to another, even without having a direct formula for the first variable in terms of the second.

Implicit differentiation requires us to differentiate both sides of the equation with respect to (\(x\)) while treating (\(y\)) as a function of (\(x\)). This means every time we differentiate a term involving (\(y\)), we multiply the derivative of that term by (\(\frac{dy}{dx}\)), the mysterious derivative we're solving for. The goal is to rearrange the resulting equation so that we isolate (\(\frac{dy}{dx}\)) and solve for it. This provides us with the rate of change of (\(y\)) in relation to (\(x\)), giving us insight into how (\(y\)) behaves as (\(x\)) changes.

Differentiating Polynomial Functions

Polynomial functions are some of the most common and smoothest functions we encounter in calculus. Differentiating them is relatively straightforward due to their simple structure: they're just sums of variables raised to whole number powers.

Our exercise involves differentiating the polynomial (\(x^4\)), which is a part of a polynomial equation. To do this, we apply the power rule, which states that for any term (\(x^n\)), its derivative with respect to (\(x\)) is (\(nx^{(n-1)}\)). Therefore, when we differentiate (\(x^4\)), we get (\(4x^3\)).

It's crucial to remember this rule when working through implicit differentiation exercises with polynomial functions, as it allows us to quickly and accurately determine the contribution of each term to the derivative.

Solving for Derivatives

Solving for derivatives in an implicit differentiation problem can sometimes feel like a treasure hunt, where (\(\frac{dy}{dx}\)) is the treasure. Once we've differentiated each term by applying the rules of differentiation to both sides of the equation, we find ourselves with an equation involving (\(\frac{dy}{dx}\)).

Our final step is to solve for (\(\frac{dy}{dx}\)). This typically involves isolating the term by using algebraic manipulation—moving all terms not involving (\(\frac{dy}{dx}\)) to the other side of the equals sign, for example. Once (\(\frac{dy}{dx}\)) stands alone, we can read off the solution, which tells us the derivative of (\(y\)) with respect to (\(x\)) in its bare form. This derivative holds the key to understanding how (\(y\)) will change in response to tiny nudges to (\(x\)), which is a cornerstone of calculus and critical for studying the dynamics of systems described by the equation at hand.