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Chapter 3: Problem 48

Calculate the derivative of the following functions. $$\cos ^{4}\left(7 x^{3}\right)$$

Short Answer

Expert verified
Question: Find the derivative of the function $y=\cos^{4}(7x^{3})$. Answer: The derivative of the function is $y'=-84x^{2}\cos^{3}(7x^{3})\sin(7x^{3})$.

Step by step solution

01

Identify the outer and inner functions

We have a composite function with two functions. The outer function is \(f(u) = \cos^{4}(u)\), and the inner function is \(g(x) = 7x^{3}\). We will find the derivative of these two functions separately.
02

Find the derivative of the outer function

First, find the derivative of \(f(u) = \cos^{4}(u)\) with respect to u. To do this, we will use the chain rule, treating \(\cos(u)\) as our new inside function, and \(u^4\) as our outer function. Let \(h(w) = w^4\), then \(f(u)=h(\cos(u))\). The derivative of the outer function is \(h'(w) = 4w^3\). Now find the derivative of the new inner function, \(\cos(u)\). The derivative of \(\cos(u)\) is \(-\sin(u)\). Now apply the chain rule: \(f'(u) = h'(\cos(u))(-\sin(u)) = 4\cos^{3}(u)(-\sin(u))= -4\cos^{3}(u)\sin(u)\).
03

Find the derivative of the inner function

Now, find the derivative of the inner function \(g(x) = 7x^{3}\). The derivative of this function, with respect to x, is \(g'(x) = 21x^{2}\).
04

Apply the chain rule

Now, apply the chain rule to find the derivative of the original composite function. The chain rule states that \({(f \circ g)'(x) = f'(g(x))g'(x)}\). Plug in the derivatives we found for \(f'(g(x))\) and \(g'(x)\) as follows: $$\frac{d}{dx}(\cos^{4}(7x^{3})) = (-4\cos^{3}(7x^{3})\sin(7x^{3}))(21x^{2})$$
05

Simplify the expression

Simplify the expression to get the final derivative: $$\frac{d}{dx}(\cos^{4}(7x^{3})) = -84x^{2}\cos^{3}(7x^{3})\sin(7x^{3})$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chain Rule
The chain rule is a fundamental tool in calculus, essential for differentiating composite functions. It's like peeling an onion layer by layer.
When we have a function within another function, we need to take the derivative of the outer function while keeping the inner function intact. Afterward, we multiply this result by the derivative of the inner function.
For example, with the outer function \(f(u) = \cos^4(u)\) and the inner function \(g(x) = 7x^3\), the chain rule guides us to first differentiate the outer layer, treating the entire inner function as a single variable, then multiply by the derivative of this inner function.
Integrating the chain rule methodically helps us to manage complex functions step-by-step with clarity and precision.
Composite Function
A composite function is a function made up of two or more simpler functions, often represented as \(f(g(x))\). They involve nesting one function inside another.
Breaking down the original problem, \( \cos^4(7x^3) \) is a composite where \(\ f(u) = \cos^4(u)\), and \(g(x) = 7x^3\). It's this combination that requires special handling with rules like the chain rule.
Understanding composite functions involves recognizing the structure: which is the outer function and which is the inner function. It鈥檚 akin to reading layers in a cake recipe, but with math functions. When you know what makes up your composite, applying calculus concepts becomes more straightforward.
Trigonometric Functions
Trigonometric functions, such as \(\sin\) and \(#\cos\), are ubiquitous in calculus, representing relationships within triangles or periodic phenomena.
They possess unique characteristics making them important in differentiation. For \(\cos(u)\), its derivative is \(-\sin(u)\).
This differentiation fact is pivotal when applying the chain rule to trigonometric composite functions. Trigonometric derivatives often involve negative sine or cosine terms, a crucial point to remember when working through derivative problems.
Visualizing these functions graphically can aid comprehension, seeing how slopes or patterns translate into these mathematically derived expressions.
Power Rule
The power rule is a straightforward yet essential differentiation rule. It states that for any function \(x^n\), the derivative is \(nx^{n-1}\). This rule simplifies finding derivatives of polynomial terms.
In our exercise, the power rule helps differentiate \(g(x) = 7x^3\), yielding \(g'(x) = 21x^2\).
Simple as it sounds, the power rule is a powerhouse for quickly reducing polynomial complexities into simpler terms.
Applying the power rule is about recognizing formats and mechanically applying the rule鈥攎ultiply by the power, reduce the power by one.

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Most popular questions from this chapter

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