Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate. $$ \frac{d}{d x}\left[\frac{1}{\sin ^{3}(\cos 2 x)}\right] $$

Step-by-Step Solution

Answer

The derivative of \( \frac{1}{\sin ^{3}(\cos 2 x)} \) with respect to \(x\) is \(-6\sin^{-2}(cos(2x))\sin(2x)\cos(cos(2x)).

Step 1: Identify the Function and its Derivative

Identify the function and its derivative by applying the chain rule. Here we have \(u = \sin^3(cos(2x))\) which is a composition of multiple functions. Take the derivative of \(u\) with respect to \(x\). The chain rule gives us \(\frac{du}{dx} = \frac{du}{dv} * \frac{dv}{dx}\), which translates to \(-3\sin^2(cos(2x)) * \cos(cos(2x)) * -2\sin(2x)\) after applying the chain rule first on \(\sin^3(v)\) and then on \(cos(2x)\) where \(v = cos(2x)\).

Step 2: Simplify

Simplify the result from the previous step. This gives \(-3\sin^2(cos(2x)) * \cos(cos(2x)) * -2\sin(2x) = 6\sin^2(cos(2x))\sin(2x)\cos(cos(2x))\).

Step 3: Multiply by the Derivative of the Denominator

Finally, realize that the original function was \(\frac{1}{\sin ^{3}(\cos 2 x)}\) which can also be written as \(\sin^{-3}(\cos 2 x)\). Thus we need to multiply the result by \(-\sin^{-4}(\cos(2x))\), the derivative of \(sin^{-3}(cos(2x))\), to get the final answer as \(-6\sin^{-2}(cos(2x))\sin(2x)\cos(cos(2x))\).

Key Concepts

Chain Rule

The chain rule is a powerful tool used in calculus to differentiate composite functions. It helps when you have functions nested within each other. To apply the chain rule, identify the outer function and the inner function. The rule can be stated as follows: if you have a function that is composed of two functions, say \( y = f(g(x)) \), the derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).
In our example, we need to differentiate \( \sin^3(\cos(2x)) \). Here:

• The outer function is \( u = \sin^3(v) \)
• The inner function is \( v = \cos(2x) \)

To find the derivative, we apply the chain rule twice - once for the sine function and once for the cosine function. It's crucial to proceed step-by-step, carefully applying the rule to maintain accuracy.

Trigonometric Functions

Trigonometric functions are fundamental in calculus and appear frequently in problems involving derivatives. These functions include sine, cosine, and tangent, among others. They have specific derivative rules:

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

In our problem, we deal with both \( \sin \) and \( \cos \), making it essential to understand how changes in these functions affect their derivatives.
For example, when differentiating \( \sin^3(\cos(2x)) \), the trigonometric rules guide us in understanding how the composition of multiple trig functions will behave. Breaking down each trigonometric component properly ensures that we can simplify the derivative correctly later.

Simplification

Simplification in calculus helps us express a complex derivative more cleanly. Once you have applied the derivative rules, you may end up with a complex expression. To simplify, you should:

• Combine like terms
• Use trigonometric identities where applicable
• Factor out common factors

In the example, initial differentiation gives us terms like \(-3\sin^2(\cos(2x)) \cdot \cos(\cos(2x)) \cdot -2\sin(2x)\). By simplifying step-by-step:

• Recognize and combine numerical coefficients (-3 \* -2 = 6)
• Consolidate similar sine and cosine terms

The simplification process allows the derivative to be presented as \( 6\sin^2(\cos(2x))\sin(2x)\cos(\cos(2x)) \), easier for further analysis or applications.

Composition of Functions

When dealing with calculus, a composition of functions occurs when one function is applied inside another. This is critical in identifying the layers within a problem and determining how the derivative should be taken.
For functions like \( \sin^3(\cos(2x)) \), identifying each layer is crucial before differentiating:

• The outermost is \( \sin^3 \), a power function of a sine function
• In the middle, there is the sine function: \( \sin(x) \)
• Innermost is the cosine function: \( \cos(2x) \)

Recognizing these nested functions allows us to apply the chain rule accurately. Be mindful of each function's role and the order of operations, which affect how derivatives are computed sequentially.
This understanding simplifies interactions between layers and helps in expressing the derivative in a clear, concise form.