Problem 5 - Calculus

Question

Find \(d^{3} y / d x^{3}\). $$ y=\sin (7 x) $$

Step-by-Step Solution

Answer

\(\frac{d^3y}{dx^3} = -343\cos(7x)\).

Step 1: Differentiate for the First Time

Differentiate the given function \(y = \sin(7x)\) with respect to \(x\). Use the chain rule, which gives \(\frac{dy}{dx} = 7\cos(7x)\).

Step 2: Differentiate the First Derivative

Differentiate \(\frac{dy}{dx} = 7\cos(7x)\) with respect to \(x\). Again, apply the chain rule: \(\frac{d^2y}{dx^2} = -49\sin(7x)\).

Step 3: Differentiate the Second Derivative

Find the third derivative \(\frac{d^3y}{dx^3}\) by differentiating \(\frac{d^2y}{dx^2} = -49\sin(7x)\). Use the chain rule again to get \(\frac{d^3y}{dx^3} = -343\cos(7x)\).

Key Concepts

Chain Rule

The Chain Rule is a fundamental tool in calculus for finding the derivative of composite functions. A composite function is when you have a function within another function, such as the sine function inside the cosine function. This rule allows us to differentiate more complex functions by breaking them down into simpler parts.
To apply the Chain Rule, imagine you have a function of the form \( y = \sin(7x) \). Here, the inner function is \( 7x \) and the outer function is \( \sin(u) \) where \( u = 7x \).
The Chain Rule states that the derivative of \( y \) with respect to \( x \) is the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function with respect to \( x \). So, the derivative becomes:

• Find the derivative of \( \sin \) which is \( \cos(u) \)
• Multiply by the derivative of \( 7x \), which is \( 7 \)

Making it \( \frac{dy}{dx} = 7 \cos(7x) \). This process demonstrates how the Chain Rule simplifies complex differentiation problems.

Higher Order Derivatives

Higher order derivatives are derivatives of a function taken multiple times. For example, if you have already differentiated a function once to get its first derivative, finding the second derivative involves differentiating the first derivative.
In similar fashion, the third derivative is found by differentiating the second derivative.
For the function \( y = \sin(7x) \), calculating the first derivative using the Chain Rule gives \( \frac{dy}{dx} = 7\cos(7x) \). Taking the derivative of this expression gives the second derivative, \( \frac{d^2y}{dx^2} = -49\sin(7x) \). Continuing this process:

• Differentiate the second derivative \( -49\sin(7x) \)
• Apply the Chain Rule again to get the third derivative
• Resulting in \( \frac{d^3y}{dx^3} = -343\cos(7x) \)

Higher order derivatives can give insights into a function's behavior, such as its concavity or points of inflection.

Calculus Tutorial

Let's break down the process of finding higher order derivatives in this tutorial. It's a step-by-step guide to understanding not just the mechanics but the principles behind calculus operations as well.
Start with a basic understanding of derivatives. A derivative is a measure of how a function changes as its input changes. When dealing with trigonometric functions like \( \sin(x) \), their derivatives are cyclic and periodic, often involving further mathematical rules such as the Chain Rule.
To conceptualize higher order derivatives:

• Consider derivatives as a way to describe the change of change.
• A first derivative describes the immediate rate of change.
• Second and third derivatives further refine how this rate itself changes.

In this calculus tutorial, practice with simple exercises like differentiating \( y = \sin(7x) \). Start by identifying the inner and outer functions, carry through with the Chain Rule, and repeat for higher orders.
By regularly engaging with such problems, calculus becomes less about solving puzzles and more about understanding changes, whether simple or complex.