Problem 27 - Thomas Calculus in SI Units

Question

Find \(d y / d x.\) $$e^{2 x}=\sin (x+3 y)$$

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = \frac{2e^{2x} - \cos(x+3y)}{3\cos(x+3y)}\)

Step 1: Differentiate Both Sides

Begin by applying implicit differentiation to both sides of the equation with respect to \(x\). Differentiating the left side, the derivative of \(e^{2x}\) with respect to \(x\) is \(2e^{2x}\).

Step 2: Differentiate the Right Side Using Chain Rule

The right side is \(\sin(x + 3y)\). Using the chain rule, its derivative is \(\cos(x + 3y) \cdot (1 + 3\frac{dy}{dx})\), where \(1\) is the derivative of \(x\) and \(3\frac{dy}{dx}\) is the derivative of \(3y\).

Step 3: Combine and Solve for \(\frac{dy}{dx}\)

Set the derivatives from Step 1 and Step 2 equal: \[ 2e^{2x} = \cos(x + 3y) \cdot (1 + 3\frac{dy}{dx}) \]Rearrange this equation to solve for \(\frac{dy}{dx}\). Distribute \(\cos(x + 3y)\) on the right side and isolate \(\frac{dy}{dx}\):\[ 2e^{2x} = \cos(x + 3y) + 3\cos(x + 3y)\frac{dy}{dx} \]Thus, \[ 3\cos(x + 3y)\frac{dy}{dx} = 2e^{2x} - \cos(x + 3y) \]

Step 4: Final Isolate \(\frac{dy}{dx}\)

Divide both sides by \(3\cos(x + 3y)\) to isolate \(\frac{dy}{dx}\):\[ \frac{dy}{dx} = \frac{2e^{2x} - \cos(x + 3y)}{3\cos(x + 3y)} \]

Key Concepts

Chain Rule

The Chain Rule is an essential concept in calculus used for differentiating composite functions. It allows you to find the derivative of functions that are composed of other functions.
Consider a simple function composed of two functions like \(f(g(x))\). According to the Chain Rule, the derivative is \(f'(g(x)) \cdot g'(x)\).
Breaking it down:

• You first differentiate the outer function \(f\) while keeping the inner function \(g(x)\) unchanged.
• Then, multiply this derivative by the derivative of the inner function \(g(x)\).

In the given exercise, the Chain Rule helps us differentiate \(\sin(x + 3y)\). Here, the outer function is \(\sin\), and its derivative is \(\cos\).
The inner function is \(x + 3y\), leading to a derivative of \((1 + 3\frac{dy}{dx})\) because \(x\) differentiates to 1 and \(3y\) to \(3\frac{dy}{dx}\). This example highlights how you recognize and skillfully apply the Chain Rule to composite functions.

Trigonometric Differentiation

Trigonometric functions like \(\sin, \cos, \tan\) and more have standard derivatives. Which makes them straightforward to work with in calculus. Knowing these derivatives is crucial for solving trigonometric differentiation problems.

Let's review the key derivatives:

• The derivative of \(\sin(x)\) is \(\cos(x)\).
• The derivative of \(\cos(x)\) is \(-\sin(x)\).
• The derivative of \(\tan(x)\) is \(\sec^2(x)\).

In this exercise, we applied these basics to the function \(\sin(x + 3y)\).
Using the derivative of \(\sin\), we obtain \(\cos(x + 3y)\). Then, as we understand with the Chain Rule, we multiply by the derivative of the inner terms \(x + 3y\).
It's a beautiful combination of basic trigonometric derivatives and the Chain Rule. This approach showcases how these formulas provide a pathway to tackling more complex functions with ease.

Exponential Functions

Exponential functions are unique, particularly due to their constant base. Which leads to certain characteristics in their differentiation. The most common form is \(e^{x}\), where \(e\) is the natural exponential base.

Let's focus on the derivatives:

• The derivative of \(e^{x}\) remains \(e^{x}\), reflecting the unchanging rate of growth.
• If the exponent is more complex, such as \(e^{2x}\), use the Chain Rule. The derivative becomes \(2e^{2x}\), multiplying by the derivative of the exponent \(2x\).

In the given exercise, the differentiation of \(e^{2x}\) is straightforward but neatly demonstrates these principles.
Using \(2e^{2x}\), derived from the power \(2x\), shows how exponential growth can be efficiently captured in calculus.
These concepts reinforce the elegance of exponential functions, vital in rich mathematical and scientific applications.