Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { If } y=\sin \left(x^{2}\right)+2 x^{3} \text {, find } d x / d y \text {. } $$

Step-by-Step Solution

Answer

\( \frac{dx}{dy} = \frac{1}{\cos(x^2) \cdot 2x + 6x^2} \).

Step 1: Identify the Expression

We are given the expression \( y = \sin(x^2) + 2x^3 \). Our goal is to find \( \frac{dx}{dy} \), the derivative of \( x \) with respect to \( y \).

Step 2: Differentiate with Respect to x

Differentiate both sides of the expression with respect to \( x \). Using the chain rule for \( \sin(x^2) \) and the power rule for \( 2x^3 \), we get:\[ \frac{dy}{dx} = \cos(x^2) \cdot 2x + 6x^2 \].

Step 3: Solve for dx/dy

Recall that \( \frac{dy}{dx} \) is the derivative of \( y \) with respect to \( x \), so to find \( \frac{dx}{dy} \), you need to take the reciprocal:\[ \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{\cos(x^2) \cdot 2x + 6x^2} \].

Key Concepts

Chain Rule

The chain rule is a powerful tool in calculus for differentiating composite functions. Think of it as a way to unwrap complex functions layer by layer. In our exercise, we are faced with differentiating \( \sin(x^2) \), which is a great example of a composite function.

Here's how the chain rule comes into play:

• Identify the outer function and the inner function. For \( \sin(x^2) \), the outer function is \( \sin(u) \) and the inner function is \( u = x^2 \).
• Differentiate the outer function with respect to the inner function: \( \cos(u) \).
• Differentiate the inner function with respect to \( x \): \( 2x \).

Finally, you multiply them together to get the derivative of the composite function: \( \cos(x^2) \cdot 2x \). By breaking it into steps, the chain rule turns tricky differentiation problems into manageable parts.

Reciprocal Derivative

The reciprocal of a derivative is another key concept, especially in implicit differentiation problems. When you have \( \frac{dy}{dx} \), which represents how \( y \) changes with \( x \), sometimes you need to find the opposite: \( \frac{dx}{dy} \).

This is where the reciprocal derivative steps in: If you know \( \frac{dy}{dx} \), finding \( \frac{dx}{dy} \) is straightforward.

• Simply take the reciprocal: \( \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} \).

In our problem, after finding \( \frac{dy}{dx} = \cos(x^2) \cdot 2x + 6x^2 \), we then find \( \frac{dx}{dy} \) by reciprocating: \( \frac{1}{\cos(x^2) \cdot 2x + 6x^2} \). This inversion is crucial in getting the derivative in the direction you need it.

Power Rule

The power rule is one of the simplest yet most important rules in differentiation. It allows us to rapidly turn powers into coefficients. For a function \( x^n \), the power rule tells us that the derivative is \( nx^{n-1} \).

This rule is straightforward and is applied to straightforward expressions like \( x^3 \). For our exercise, we apply the power rule to \( 2x^3 \).

• Bring down the power as a coefficient: 3.
• Multiply it by the existing coefficient: 2.
• Decrease the power by one: \( x^{2} \).

So, differentiating \( 2x^3 \) using the power rule gives us \( 6x^2 \). This process quickly returns a derivative for polynomial terms, thus efficiently tackling part of our broader differentiation problem.