Problem 29 - Thomas Calculus

Question

Use implicit differentiation to find \(d y / d x\) in Exercises \(19-32\) $$ x+\tan (x y)=0 $$

Step-by-Step Solution

Answer

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

Step 1: Differentiate Both Sides

We start by differentiating both sides of the equation with respect to \(x\). The given equation is \(x + \tan(xy) = 0\). This gives us the derivatives:\[\frac{d}{dx}(x) + \frac{d}{dx}(\tan(xy)) = \frac{d}{dx}(0).\]

Step 2: Differentiate x

The derivative of \(x\) with respect to \(x\) is 1. Therefore, we have: \[1 + \frac{d}{dx}(\tan(xy)) = 0.\]

Step 3: Differentiate tan(xy) using Chain Rule

Apply the chain rule to differentiate \(\tan(xy)\). The derivative of \(\tan u\) where \(u = xy\) is \(\sec^2(u) \cdot \frac{du}{dx}\). First, differentiate \(xy\) using the product rule to get \( \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \). Thus, the derivative of \(\tan(xy)\) becomes \(\sec^2(xy) (x \frac{dy}{dx} + y)\).

Step 4: Set up the Equation After Differentiation

Substitute back the differentiated terms into the equation: \[1 + \sec^2(xy) \cdot (x \frac{dy}{dx} + y) = 0.\]

Step 5: Solve for \(\frac{dy}{dx}\)

Rearrange to solve for \(\frac{dy}{dx}\):\[\sec^2(xy) \cdot x \frac{dy}{dx} = -1 - \sec^2(xy) \cdot y.\]Now, isolate \(\frac{dy}{dx}\):\[x \sec^2(xy) \frac{dy}{dx} = -1 - y \sec^2(xy)\;\Rightarrow\;\frac{dy}{dx} = \frac{-1 - y \sec^2(xy)}{x \sec^2(xy)}.\]

Step 6: Simplify the Expression for \(\frac{dy}{dx}\)

Finally, simplify the expression if possible. The equation for \(\frac{dy}{dx}\) is:\[\frac{dy}{dx} = \frac{-1}{x \sec^2(xy)} - \frac{y}{x}.\] This simplifies the understanding of how \(\frac{dy}{dx}\) changes with respect to \(x\) and \(y\).

Key Concepts

Chain Rule

When we deal with implicit differentiation, particularly for functions of multiple variables like in this example, the chain rule becomes invaluable. It helps us differentiate composite functions. In the given function \( \tan(xy) \), think of \( xy \) as a single unit or \( u \). The chain rule tells us how to differentiate \( \tan(u) \) with respect to \( x \) when \( u \) is itself a function of \( x \).
The chain rule formula is:

• \( \frac{d}{dx}[\tan(u)] = \sec^2(u) \cdot \frac{du}{dx} \)

Applying this, we first find the derivative of \( xy \) with respect to \( x \) (for \( \frac{du}{dx} \)), which we get using the product rule. The chain rule is essential here, allowing us to combine multiple rules efficiently to find derivatives of layered functions.

Product Rule

The product rule is crucial when differentiating products of two or more functions. For the term \( xy \) in our problem, where both \( x \) and \( y \) are variables, the product rule is applied. The product rule states:

• \( \frac{d}{dx}[uv] = u \frac{dv}{dx} + v \frac{du}{dx} \)

For \( xy \), our \( u \) is \( x \) and \( v \) is \( y \). The differentiation gives:

• \( \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \)

Using the product rule allows us to find derivatives when both multiplicands change with respect to \( x \), which is common in implicit differentiation scenarios.

Trigonometric Derivatives

Trigonometric derivatives occur frequently in calculus problems, especially when functions are embedded within trigonometric expressions. The derivative of \( \tan \) involves the secant function:

• \( \frac{d}{dx}[\tan(x)] = \sec^2(x) \)

In our example, the argument of the \( \tan \) function is the product \( xy \). This makes chain rules and trigonometric derivatives intertwined. We end up differentiating \( \tan(xy) \), using the chain rule to incorporate the derivative of the product \( xy \).
Understanding trigonometric derivatives is key to solving implicit differentiation problems involving trigonometric functions.

Differentiation Steps

**Understanding Differentiation Steps**The process of solving a differentiation problem of this complexity requires a sequential approach:

• Step 1: Differentiate both sides of the equation with respect to \( x \). For example, in \( x + \tan(xy) = 0 \), we differentiate to get \( 1 + \frac{d}{dx}[\tan(xy)] = 0 \).
• Step 2: Use the chain rule on the trigonometric function \( \tan(xy) \) to find its derivative.
• Step 3: Apply the product rule inside the chain rule differentiation process on \( xy \) to find its derivative.
• Step 4: Plug the derivatives back into the original equation. Simplify what's possible.
• Step 5: Solve for \( \frac{dy}{dx} \) as the problem requires.
• Step 6: Simplify the expression if necessary, to have a clear understanding of how \( y \) changes with \( x \).

By following these steps methodically, we ensure that every aspect of the differentiation is thorough and correct. This structure is essential for tackling implicit differentiation problems efficiently.