Question

Find \(\frac{d y}{d x}\) using implicit differentiation. $$\frac{x^{2}+y}{x+y^{2}}=17$$

Short answer

\( \frac{dy}{dx} = \frac{17 - 2x}{1 - 34y} \)

Step-by-step solution

Rewrite the Equation

Start with the given equation \( \frac{x^{2}+y}{x+y^{2}}=17 \). Apply implicit differentiation, where both \( x \) and \( y \) are treated as functions of \( x \). Begin by rewriting the denominator as \( (x+y^2) \cdot 17 \) to create an equivalent multiplication form: \( x^2 + y = 17(x + y^2) \).

Expand and Differentiate the Equation

Expand the right side: \( x^2 + y = 17x + 17y^2 \). Differentiate both sides of this equation with respect to \( x \). Recall that when differentiating \( y \) with respect to \( x \), you should multiply by \( \frac{dy}{dx} \) (chain rule).

Differentiate the Left Side

Differentiate \( x^2 + y \) with respect to \( x \):\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y) = 2x + \frac{dy}{dx} \]

Differentiate the Right Side

Differentiate \( 17x + 17y^2 \) with respect to \( x \):\[ \frac{d}{dx}(17x) + \frac{d}{dx}(17y^2) = 17 + 34y\frac{dy}{dx} \]

Set Derivatives Equal

Set the derivatives obtained from both sides of the expanded equation equal to each other:\[ 2x + \frac{dy}{dx} = 17 + 34y \frac{dy}{dx} \]

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

Rearrange the equation to solve for \( \frac{dy}{dx} \):\[ \frac{dy}{dx} - 34y \frac{dy}{dx} = 17 - 2x \]Factor \( \frac{dy}{dx} \):\[ \frac{dy}{dx}(1 - 34y) = 17 - 2x \]Finally, solve for \( \frac{dy}{dx} \):\[ \frac{dy}{dx} = \frac{17 - 2x}{1 - 34y} \]

Key concepts

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. If you have a function within another function, the chain rule helps you find the derivative. This is especially useful when dealing with implicit differentiation, where you treat variables as functions of another variable, often time or in this case, x.

• When differentiating implicitly, always take the derivative of the outer function first.
• You then multiply by the derivative of the inner function. This is the part where the chain comes into play.
• In this exercise, when encountering the term with y, you treat y as a function of x. So, when you differentiate y, you multiply by \( \frac{dy}{dx} \).

This step adds an extra term to your differentiation process each time you differntiate something involving y. Remember this key rule and you’ll be able to tackle any similar problem with confidence.

Differentiation Techniques

Differentiation is the process of finding the derivative, which is essentially the rate of change of a function. Different techniques apply based on the form of the function you're dealing with. Implicit differentiation is one such technique, often used when it is difficult or impossible to solve for one variable in terms of another.

• In implicit differentiation, treat all variables as functions of x. Even if you can't explicitly solve for y, you still differentiate as if y is a function of x.
• Apply the product rule and chain rule as needed. For instance, notice how the terms are expanded in the equation \( x^2 + y = 17(x + y^2) \). This allows you to apply these rules effectively.
• Each time you differentiate a term involving y, remember to multiply by \( \frac{dy}{dx} \) as a result of the chain rule.

Mastering these techniques will improve your problem-solving efficiency when dealing with more complex calculus problems.

Solving Equations

After you've differentiated both sides of the equation during implicit differentiation, the next crucial step is solving for \( \frac{dy}{dx} \). This involves rearranging the equation to isolate this term, which can often be the most challenging part.

• First, set the derivatives equal to each other. This gives you an equation involving \( \frac{dy}{dx} \).
• Carefully rearrange the terms, gathering all instances of \( \frac{dy}{dx} \) on one side of the equation.
• You may need to factor the expression to solve for \( \frac{dy}{dx} \). As shown, factor out \( \frac{dy}{dx} \) to simplify the equation.
• Finally, divide and solve for \( \frac{dy}{dx} \), ensuring you simplify the expression as much as possible.

By following these steps, you'll derive the formula \( \frac{dy}{dx} = \frac{17 - 2x}{1 - 34y} \), which gives the rate of change of y with respect to x, as required by the original problem. Consistent practice with solving these kinds of equations will sharpen your calculus skills significantly.