Question

Calculate the derivative of the following functions. $$y=\cos ^{4} \theta+\sin ^{4} \theta$$

Short answer

The derivative of \(y = \cos^4\theta + \sin^4\theta\) with respect to \(\theta\) is: $$y' = \frac{dy}{d\theta} = -4\cos^3\theta \sin\theta + 4\sin^3\theta \cos\theta$$

Step-by-step solution

Identify the functions to be differentiated

We are given a function: $$y=\cos ^{4} \theta+\sin ^{4} \theta$$ We need to find the derivative \(y'=\frac{dy}{d\theta}\). To do this, we will apply the chain rule for differentiation since the function is a combination of two basic trigonometric functions raised to the power of 4.

Use the chain rule

The chain rule for calculating a derivative states that if we have a composite function: $$h(x) = f(g(x))$$ Then its derivative is given by: $$h'(x) = f'(g(x)) \cdot g'(x)$$ In our problem, we can identify two sub-functions: the outer functions are the powers of 4, and the inner functions are the trigonometric functions. Thus, we have: $$f_1(\theta) = \cos^4 \theta \qquad g_1(\theta) = \cos \theta$$ $$f_2(\theta) =\sin^4 \theta \qquad g_2(\theta) = \sin \theta$$ Now, let's differentiate each element separately and apply the chain rule.

Differentiate the outer functions

We differentiate the outer functions \(f_1(\theta)\) and \(f_2(\theta)\): $$f_1'(\theta) = \frac{d(\cos^4\theta)}{d(\cos \theta)} = 4\cos^3\theta$$ $$f_2'(\theta) = \frac{d(\sin^4\theta)}{d(\sin \theta)} = 4\sin^3\theta$$

Differentiate the inner functions

Next, differentiate the inner functions \(g_1(\theta)\) and \(g_2(\theta)\): $$g_1'(\theta) = \frac{d(\cos \theta)}{d\theta} = -\sin \theta$$ $$g_2'(\theta) = \frac{d(\sin \theta)}{d\theta} = \cos \theta$$

Apply the chain rule

Now let's apply the chain rule to find the derivative of the entire function: $$\frac{dy}{d\theta} = f_1'(g_1(\theta)) \cdot g_1'(\theta) + f_2'(g_2(\theta)) \cdot g_2'(\theta)$$ Substitute the derivatives and simplify: $$\frac{dy}{d\theta} = (4\cos^3\theta) \cdot (-\sin \theta) + (4\sin^3\theta) \cdot (\cos \theta)$$ $$\frac{dy}{d\theta} = -4\cos^3\theta \sin\theta + 4\sin^3\theta \cos\theta$$ The derivative of the given function is: $$y' = \frac{dy}{d\theta} = -4\cos^3\theta \sin\theta + 4\sin^3\theta \cos\theta$$

Key concepts

Chain Rule

The Chain Rule is a powerful technique in calculus used for finding the derivative of composite functions. In simpler terms, a composite function is a function within another function. Calculating its derivative isn't as straightforward as a simple differentiation because of this nested structure.

To use the Chain Rule, we identify two parts in the composite function: the inner function and the outer function. For instance, in the given exercise, both parts of the function \[y = \cos^4 \theta + \sin^4 \theta\]are such composite functions. Here,

• The outer function is the power function of 4,
• The inner functions include \(\cos \theta\) and \(\sin \theta\).

The Chain Rule states:\[\text{If } h(x) = f(g(x)), \text{ then } h'(x) = f'(g(x)) \cdot g'(x)\]Breaking this down makes the problem manageable, as you can differentiate each component separately and combine them according to this rule, which then gives you the derivative of the entire composite function.

Trigonometric Derivatives

Trigonometric derivatives are foundational expressions in calculus. Most basic trigonometric functions have well-known derivatives that are vital for solving derivative-related problems effectively. In the exercise you're working with, the derivatives of the basic trigonometric functions \(\cos \theta\) and \(\sin \theta\) are crucial.

Here's a quick refresher of the derivatives of these functions:

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

When combined with the Chain Rule, these derivatives help in determining the rate of change of more complex functions. Knowing these basic derivatives allows you to swiftly apply the Chain Rule when nested trigonometric functions are involved, as in our main exercise.

Calculus Problem Solving

Calculus problem solving involves breaking down a complex mathematical problem into more manageable parts. This exercise demonstrates how several calculus concepts can come together to solve a derivative problem. The goal is to find the rate of change of the given function.

Here's a systematic approach you can follow when tackling similar problems:

• **Identify the parts of the function**: Understand what parts of your function are nested and determine what methods, like the Chain Rule, might be applied.
• **Derive step-by-step**: Differentiate each part of the composite function separately — start with inner and then outer functions — simplifying at each stage.
• **Combine results**: Use the Chain Rule to bring together the derivatives of your composite functions in a way that calculates the derivative of the entire expression.
• **Refine and simplify**: Simplify your derivative to get it into the cleanest form possible.

By following these steps and understanding each concept's role in the process, calculus problems become easier to approach and solve. Remember, practice makes perfect, and the more you work through these steps, the more intuitive they will become.