Problem 10 - Calculus for Biology and Medicine

Question

Find \(\frac{d y}{d x}\) if \((\sin x+\cos y) x^{2}=0\)

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \frac{x \cos x + 2\sin x + 2\cos y}{x \sin y} \)

Step 1: Identify the Problem

We need to find the derivative \( \frac{d y}{d x} \) of the implicit function \( (\sin x + \cos y) x^{2} = 0 \). Since there is no explicit solution for \( y \), we will use implicit differentiation.

Step 2: Differentiate Both Sides with Respect to x

Apply implicit differentiation to both sides of the equation. The equation is \( (\sin x + \cos y) x^{2} = 0 \). The derivative of 0 is 0, so we focus on differentiating the left-hand side.

Step 3: Use the Product Rule

The left side of the equation is a product of two functions: \( (\sin x + \cos y) \) and \( x^{2} \). Use the product rule, which states: \( (f \,g)' = f' \, g + f \, g' \).

Step 4: Differentiate \( \sin x + \cos y \)

Differentiating \( \sin x \) with respect to \( x \) gives \( \cos x \), and differentiating \( \cos y \) with respect to \( x \) using the chain rule gives \(-\sin y \cdot \frac{dy}{dx} \).

Step 5: Complete the Product Rule

The derivative of \( (\sin x + \cos y) x^{2} \) is \( (\cos x - \sin y \frac{dy}{dx}) x^2 + (\sin x + \cos y) (2x) = 0 \).

Step 6: Simplify and Solve for \( \frac{d y}{d x} \)

Distribute and collect all terms:\[ x^2 \cos x - x^2 \sin y \frac{dy}{dx} + 2x \sin x + 2x \cos y = 0. \] Rearrange to solve for \( \frac{dy}{dx} \):\[ -x^2 \sin y \frac{dy}{dx} = -x^2 \cos x - 2x \sin x - 2x \cos y. \] Divide through by \(-x^2 \sin y \):\[ \frac{dy}{dx} = \frac{x \cos x + 2\sin x + 2\cos y}{x \sin y}. \]

Key Concepts

Product Rule

When differentiating products of functions, it's crucial to apply the Product Rule. This rule simplifies the process of finding derivatives and is particularly useful in implicit differentiation problems where functions of different variables are multiplied together.
For two functions, say \( f(x) \) and \( g(x) \), the Product Rule states that the derivative is:

• \( (f \cdot g)' = f' \cdot g + f \cdot g' \)

In our exercise, the expression \( (\sin x + \cos y) x^{2} = 0 \) involves a product of \( (\sin x + \cos y) \) and \( x^2 \). By using the Product Rule, we get:

• Differentiate \( (\sin x + \cos y) \) and multiply it by \( x^2 \)
• Then, differentiate \( x^2 \) and multiply it by \( (\sin x + \cos y) \)

This approach helps us break down complex derivatives step by step, making them easier to solve.

Chain Rule

Understanding the Chain Rule is key when dealing with composite functions in differentiation, such as those in our problem. The Chain Rule helps find the derivative of a composition of two functions.
When you have a function \( h(x) = f(g(x)) \), the Chain Rule states:

• \( h'(x) = f'(g(x)) \cdot g'(x) \)

In this implicit differentiation example, we differentiate \( \cos y \) with respect to \( x \). Since \( y \) is a function of \( x \), applying the Chain Rule looks like this:

• Differentiating \( \cos y \) gives \(-\sin y\)
• Then multiply by \( \frac{dy}{dx} \) because \( y \) depends on \( x \)

Thus, incorporating the Chain Rule allows us to handle terms where one variable is embedded within another. This is crucial for solving implicitly defined equations.

Derivative Calculation

Calculating derivatives, especially in implicit differentiation, involves decomposing each term correctly by applying relevant rules. The overall objective is to find \( \frac{d y}{d x} \), which measures how much \( y \) changes with a small change in \( x \).
For this problem:

• Start by differentiating both sides of the equation. The right side is zero, which simplifies our task.
• Then, using the Product Rule and Chain Rule, compute derivatives for the left side.

Upon simplifying, you gather all terms involving \( \frac{dy}{dx} \) on one side. The other terms are the numerical derivatives calculated earlier.
Rearrange the equation to isolate \( \frac{dy}{dx} \):

• For instance, shift terms around to get \( -x^2 \sin y \frac{dy}{dx} = -x^2 \cos x - 2x \sin x - 2x \cos y \)
• Finally, divide both sides by \( -x^2 \sin y \) to solve for \( \frac{dy}{dx} \).

This sequence allows us to find the implicit derivative correctly, ensuring we account for all variables, constants, and how they interact through differentiation rules.