Question

Apply the Chain Rule more than once to find the indicated derivative. \(D_{\theta}\left[\cos ^{4}\left(\sin \theta^{2}\right)\right]\)

Short answer

The derivative is \(-8\theta \sin(\theta^2) \cos(\theta^2) \cos^3(\sin(\theta^2))\)."

Step-by-step solution

Identify the outer function

We need to differentiate \( \cos^4(u) \), where \( u = \sin(\theta^2) \). The outer function is expressed as \( f(u) = (\cos(u))^4 \). We can differentiate this using the Chain Rule.

Apply the Chain Rule to the outer function

Differentiate \( (\cos(u))^4 \) with respect to \( u \). This gives: \[ f'(u) = 4(\cos(u))^3 \cdot (-\sin(u))\]Thus, the derivative of the outer function is \( -4\sin(u)\cos^3(u) \).

Identify and differentiate the middle function

The middle function is \( u = \sin(v) \) where \( v = \theta^2 \). Differentiate \( \sin(v) \) with respect to \( v \):\[ g'(v) = \cos(v)\]So, we replace \( u \) with \( \sin(\theta^2) \) to find the derivative of the middle function: \( \cos(\theta^2) \).

Differentiate the innermost function

The innermost function is \( v = \theta^2 \). Differentiate \( \theta^2 \) with respect to \( \theta \):\[ h'(\theta) = 2\theta\]

Chain all the derivatives together

According to the Chain Rule, the derivative \( \frac{d}{d\theta} \left[\cos^4(\sin(\theta^2))\right] \) is:\[ -4\sin(\theta^2) \cdot \cos^3(\sin(\theta^2)) \cdot \cos(\theta^2) \cdot 2\theta\]This simplifies to:\[ -8\theta \sin(\theta^2) \cos(\theta^2) \cos^3(\sin(\theta^2))\]

Key concepts

Derivative

A derivative tells us how a function changes as its input changes. If you think of a function as a machine, the derivative describes how the output of that machine changes when you turn the dials on the input slightly. In simpler terms, it's like finding the slope of a line that is tangent to a curve at any point. This slope helps us understand the function's rate of change at that particular point.

The process of finding a derivative is called differentiation. Not every function is as straightforward to differentiate as simple polynomials or lines. Some functions, like those involving trigonometric or composite functions, require more advanced techniques, such as the Chain Rule. By finding derivatives, we can solve a wide range of real-world problems involving rates of change, like velocity and other changes in motion.

Trigonometric Functions

Trigonometric functions, such as sine \( \sin \), cosine \( \cos \), and tangent \( \tan \), are fundamental in the study of triangles and modeling periodic phenomena like waves. These functions provide powerful ways to relate angles to the ratios of sides in right-angled triangles.

When differentiating trigonometric functions, it's important to remember their specific derivative rules:

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

These rules are crucial for effectively handling problems where these functions present themselves, as seen in the original exercise where we have \( \sin(\theta^2) \) and \( \cos(\sin(\theta^2)) \). Differentiating these functions requires careful application of these derivative rules, often in combination with the Chain Rule for composite functions.

Composite Functions

Composite functions are functions made by combining two or more functions. For example, if you have a function \( f(x) \) and you input another function \( g(x) \) into it, creating \( f(g(x)) \), you now have a composite function. These functions can involve layers of functions, like nesting a sine function inside a cosine function and then raising it to a power, as seen in the exercise.

When differentiating composite functions, the Chain Rule becomes a vital tool. The Chain Rule states that the derivative of a composite function \( f(g(x)) \) is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Mathematically, it is expressed as:

• \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)

Applying the Chain Rule more than once is often necessary when dealing with multiple nested functions, as in the derivatives like \( \cos^4(\sin(\theta^2)) \). Through careful identification and differentiation of each layer, the overall derivative can be effectively found, just like piecing together a puzzle step by step.