Problem 21 - Technical Mathematics with Calculus

Question

Find \(d y / d x\). (Treat \(a\) and \(r\) as constants.) $$y+y^{3}=x+x^{3}$$

Step-by-Step Solution

Answer

The derivative \(\frac{dy}{dx}\) of the equation \(y+y^{3}=x+x^{3}\) is found to be \(\frac{dy}{dx}=\frac{1+3x^2}{1+3y^2}\).

Step 1: Differentiate Both Sides

Differentiate both sides of the equation with respect to x, using the power rule for differentiation. Remember to treat all constants as zero when differentiating with respect to x, and any term with y will require the use of the chain rule since it is a function of x.

Step 2: Apply the Chain Rule

When differentiating the terms involving y, apply the chain rule. For each term with y, we treat y as a function of x and multiply by the derivative of y with respect to x (denoted as dy/dx).

Step 3: Rearrange and Solve for dy/dx

Once both sides are differentiated, collect all the terms involving dy/dx on one side of the equals sign, and move the rest to the other side. Finally, factor out dy/dx and solve for it by dividing both sides by the remaining coefficient.

Key Concepts

Power Rule for Differentiation

The power rule for differentiation is a basic yet powerful tool in calculus. It states that if you have a function of the form \( f(x) = x^n \), where \( n \) is any real number, the derivative of that function with respect to \( x \) is \( f'(x) = n \times x^{n-1} \). Applied to an equation like \( y + y^3 = x + x^3 \), we differentiate each term that is a power of \( x \) or \( y \) using this rule. In applying the power rule, remember that the derivative of a constant is zero, and that for terms involving \( y \), implicit differentiation is needed.

To give an example, when differentiating \( x^3 \), we bring down the exponent as a coefficient and reduce the exponent by 1 to get \( 3x^2 \). On the other hand, differentiating \( y^3 \) with respect to \( x \) requires the chain rule to account for \( y \) being a function of \( x \), which we'll explore in the next section.

Chain Rule

The chain rule is an essential concept in calculus and is particularly important when performing implicit differentiation. When a function \( y \) is dependent on \( x \), and you have \( y \) raised to a power as in \( y^3 \), you cannot directly apply the power rule without considering \( y \)'s dependence on \( x \). Here, the chain rule comes into play: you first apply the power rule as if \( y \) were an independent variable, and then multiply by the derivative of \( y \) with respect to \( x \), or \( \frac{dy}{dx} \).

Implicit differentiation involving the chain rule transforms the differentiation of \( y^3 \) with respect to \( x \) into \( 3y^2 \cdot \frac{dy}{dx} \), since \( \frac{dy}{dx} \) is the derivative of \( y \) in relation to \( x \). When applying the chain rule, it keeps the relationship between variables intact and permits the solution of more complex differentiation problems.

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It has two fundamental branches: differential calculus concerning the rate of change and slopes of curves, and integral calculus which covers the accumulation of quantities and the areas under and between curves. When we perform operations such as differentiation, we are working within the realm of differential calculus.

The problem \( y+y^3=x+x^3 \) is an example where calculus is applied to find the relationship between changing quantities. Implicit differentiation, the topic of this exercise, is a technique in calculus used to find the derivative of a function that is not explicitly solved for one variable in terms of another. Through calculus, we can not only solve for derivatives but gain an understanding of the behavior of equations and the functions they represent.

Solving Differential Equations

Solving differential equations is a critical application of calculus. A differential equation involves unknown functions and their derivatives, and the goal is to find the function that satisfies the given relationship. In the equation \( y + y^3 = x + x^3 \), after differentiating both sides and applying the chain rule, we arrive at a differential equation involving \( \frac{dy}{dx} \).

The next step to solve this is to collect all the terms containing \( \frac{dy}{dx} \) and other terms separately which then allows us to isolate \( \frac{dy}{dx} \) and solve for it. This manipulation simplifies the equation, producing a formula for the derivative of \( y \) with respect to \( x \), thus revealing how the rate of change of \( y \) is related to \( x \). This approach is not just about finding a numeric answer but understanding the underlying relationships and changes between variables in an equation.