Question

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

Short answer

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

Step-by-step solution

Identify Outer and Inner Functions

First, recognize that this problem requires the chain rule because the function is composed of an outer function and an inner function. The outer function is $$\cos ^{4}(u)$$, and the inner function is $$u=7x^3$$.

Differentiate the Outer Function

Next, take the derivative of the outer function with respect to its argument (u), treating u as a variable. Let $$y=\cos^4(u)$$, so the derivative can be calculated as: $$\frac{dy}{du}=4\cos^3(u) \cdot -\sin(u)$$. This is derived using the power rule and the chain rule on the cosine part of the outer function.

Differentiate the Inner Function

Now, differentiate the inner function (u) with respect to x: $$\frac{du}{dx} = \frac{d}{dx}(7x^3) = 21x^2$$.

Apply Chain Rule to Find Derivative

Finally, apply the chain rule to compute the derivative by multiplying the derivatives obtained in steps 2 and 3: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$. Substitute the expressions found in steps 2 and 3 in this equation and simplify the expression: $$\frac{dy}{dx} = (4\cos^3(u) \cdot -\sin(u))\cdot(21x^2) $$

Substitute Back the Inner Function

Now substitute the inner function (u = 7x^3) back into the expression, yielding the final answer for the derivative: $$\frac{dy}{dx} = (4\cos^3(7x^3) \cdot -\sin(7x^3))\cdot(21x^2) $$ $$\frac{dy}{dx} = -84x^2\cos^3(7x^3)\sin(7x^3)$$.

Key concepts

Chain Rule

The chain rule is an essential concept in calculus for finding the derivative of a composite function. A composite function is one which involves two or more functions nested within each other, like a function inside another function. The chain rule provides a way to handle such scenarios by taking the derivative of the outer function and the inner function separately, and then combining them.

Here's how the chain rule works: If you have a function of the form \(y = f(g(x))\), you apply the chain rule by first taking the derivative of the outer function \(f\) with respect to the inner function \(g(x)\), and multiply it by the derivative of the inner function \(g\) with respect to \(x\). This can be summarized by:\[ \frac{dy}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx} \]

In our specific example, the outer function is \(f(u) = \cos^4(u)\) and the inner function is \(g(x) = 7x^3\). This means the chain rule helps us find the derivative by focusing first on \(\cos^4(u)\) with respect to \(u\), and then \(7x^3\) with respect to \(x\). When correctly applied, it simplifies the process of differentiation by breaking it down into more manageable parts.

Power Rule

The power rule is one of the simplest and most frequently used rules in calculus for differentiation. It states that if you have a function \(y = x^n\), where \(n\) is any real number, the derivative is \(\frac{dy}{dx} = nx^{n-1}\). This rule allows you to quickly and efficiently find the derivative of any function raised to a power.

In this exercise, the power rule was used to differentiate both the expression \(\cos^4(u)\) during the differentiation of the outer function, and the expression \(7x^3\) for the inner function. For example:

• When differentiating \(\cos^4(u)\), the power rule helps us derive \(4\cos^3(u)\), interpreting \(\cos(u)\) as \(u\) raised to the fourth power.
• For the inner function, applying the power rule to \(7x^3\), we get \(21x^2\), which follows naturally from the formula \(\frac{d}{dx}(x^n) = nx^{n-1}\).

The power rule significantly reduces computation complexity, enabling us to handle exponential expressions with ease.

Differentiation

Differentiation is a fundamental process in calculus used to find the rate at which a function is changing instantaneously. This involves calculating the derivative, which provides the slope of the tangent line to the function at any given point.

In the context of our example, differentiation is used to find the derivative of \(y = \cos^4(7x^3)\). Each step in the differentiation process applies various rules, ensuring the derivative of each part of the function contributes to the final answer. By differentiating both the inner and outer functions, we accurately find \(\frac{dy}{dx}\), which represents how \(y\) changes with respect to \(x\).

Understanding differentiation requires mastering rules like the chain rule and power rule, which allow you to break down complex expressions into simpler parts. In essence, differentiation is one of the primary tools mathematicians use to analyze and interpret the behavior of functions in dynamic systems. It helps you predict how small changes in one variable can affect another, which is crucial in fields ranging from physics to economics.