Problem 13 - Basic Engineering Mathematics

Question

Find the differential coefficient of \(y=7 \sin 2 x-3 \cos 4 x\)

Step-by-Step Solution

Answer

The derivative is \(14 \cos 2x + 12 \sin 4x\).

Step 1: Differentiate the First Term

The first term is \(7 \sin 2x\). The derivative of \(\sin x\) is \(\cos x\), so for \(\sin 2x\), we apply the chain rule: differentiate \(\sin\) to get \(\cos 2x\) and then multiply by the derivative of \(2x\), which is 2. Therefore, the derivative of \(7 \sin 2x\) is \(7 \times 2 \times \cos 2x = 14 \cos 2x\).

Step 2: Differentiate the Second Term

The second term is \(-3 \cos 4x\). The derivative of \(\cos x\) is \(-\sin x\), so for \(\cos 4x\), we apply the chain rule: differentiate \(\cos\) to get \(-\sin 4x\) and then multiply by the derivative of \(4x\), which is 4. Therefore, the derivative of \(-3 \cos 4x\) is \(-3 \times (-4) \times \sin 4x = 12 \sin 4x\).

Step 3: Combine Results

Now, combine the derivatives of both terms: the derivative of \(y = 7 \sin 2x - 3 \cos 4x\) is \(14 \cos 2x + 12 \sin 4x\).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus that allows us to find the derivative of composite functions.
A composite function combines two or more functions into one, such as \((7 \sin 2x)\) or \((-3 \cos 4x)\), where you have the basic trigonometric functions \(\sin\) and \(\cos\) applied to more complex expressions like \(2x\) and \(4x\).
Here’s the essence of the chain rule:

• First, differentiate the outer function, keeping the inside function unchanged.
• Then, multiply that derivative by the derivative of the inner function.

In our exercise, when differentiating \(7 \sin 2x\), you first differentiate \(\sin 2x\) to get \(\cos 2x\), then multiply by the derivative of \(2x\), which is \(2\).
This gives us \(14 \cos 2x\) for the first term.
Similarly, for \(-3 \cos 4x\), differentiate \(\cos 4x\) to get \(-\sin 4x\) and multiply by the derivative of \(4x\), \(4\), resulting in \(12 \sin 4x\).
The chain rule simplifies handling nested functions in derivatives.
Understanding it is crucial for calculus success.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are cornerstone functions in mathematics, especially in calculus.
These functions frequently appear in physics and engineering due to their properties and periodic nature.
For differentiation,

• the sine function, \(\sin x\), is differentiated to become \(\cos x\).
• the cosine function, \(\cos x\), is differentiated to become \(-\sin x\).

Hence, the operations involved are straightforward but require precision.
In our exercise, the expressions involve composite arguments like \(2x\) and \(4x\).
This means using these basic derivative rules alongside the chain rule.
Each trigonometric function has specific properties, influencing their derivatives, such as periodicity and symmetry.
The correct application of these rules is vital when dealing with more complex trigonometric expressions.

Derivative of Sine and Cosine

Understanding the derivatives of sine and cosine functions is essential for calculus.
They form a base for many more complex operations and concepts.

• The derivative of \(\sin x\) is \(\cos x\).
• The derivative of \(\cos x\) is \(-\sin x\).

These derivative rules are crucial because they demonstrate how closely the functions are related through differentiation.
In this exercise, these basic derivatives are applied in a slightly more complex scenario involving the chain rule.
For \(\sin 2x\), the derivative \(\cos 2x\) results from the chain rule, multiplied by the derivative of \(2x\).
Similarly, for \(\cos 4x\), the derivative \(-\sin 4x\) is adjusted by the derivative of \(4x\).
Therefore, differentiating such terms quickly becomes intuitive once you understand these foundational derivative rules.Overall, mastering these fundamental rules is essential for solving more advanced calculus problems efficiently.