Problem 220 - Calculus Volume 1

Question

For the following exercise, a. decompose each function in the form \(y=f(u)\) and \(u=g(x),\) and b. find \(\frac{d y}{d x}\) as a function of \(x\). $$ y=(3 x-2)^{6} $$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = 18(3x - 2)^5 \)

Step 1: Decompose the Function

First, we need to express the given function in the form \( y = f(u) \) and \( u = g(x) \). In the given function \( y = (3x - 2)^6 \), identify the inner function \( u = g(x) = 3x - 2 \). Thus, the function becomes \( y = f(u) = u^6 \).

Step 2: Differentiate f with respect to u

Differentiate \( y = u^6 \) with respect to \( u \). The derivative of \( u^6 \) is \( \frac{dy}{du} = 6u^5 \).

Step 3: Differentiate g with respect to x

Now, differentiate \( u = 3x - 2 \) with respect to \( x \). The derivative is \( \frac{du}{dx} = 3 \).

Step 4: Apply the Chain Rule

According to the chain rule, \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \). Substitute the values from Step 2 and Step 3. \[ \frac{dy}{dx} = 6u^5 \times 3 \] \[ \frac{dy}{dx} = 18u^5 \]

Step 5: Express dy/dx as a Function of x

Substitute \( u = 3x - 2 \) back into the expression for \( \frac{dy}{dx} \). \[ \frac{dy}{dx} = 18(3x - 2)^5 \]

Key Concepts

Chain Rule

The chain rule is an essential concept in calculus used to differentiate composite functions. When you have a function nested within another, like our example where a polynomial is raised to a power, the chain rule lets us find the derivative efficiently. To apply the chain rule:

• First, identify the outer function and the inner function; typically, the outer function operates on the result of the inner function.
• Differentiation is then broken down: start with the outer function, taking it with respect to the inner variable, and then move to the inner function, taking it with respect to the actual variable you are interested in.
• Multiply these derivatives to find the overall derivative.

In a nutshell, think of the chain rule as peeling the layers of an onion. You tackle the outer layers first, then move inward. This structured approach is particularly helpful for handling complex functions in calculus.

Derivative

The derivative measures how a function changes as its input changes. It's a core concept in calculus, providing the slope of the tangent line to the graph of a function at any point. For polynomials, differentiation follows simple rules:

• The power rule states: if you have a term like \(x^n\), its derivative is \(nx^{n-1}\). So, you bring the power down as a coefficient and reduce the power by one.
• For constants like \(c\), the derivative is \(0\) because constants don’t change, so they don’t have a slope.

These basics allow us to compute derivatives efficiently. Whether you're dealing with a single variable or more complex equations, grasping the concept of derivatives is vital to solving calculus problems effectively.

Function Decomposition

Function decomposition involves breaking down complex functions into simpler parts, making them easier to work with, especially when applying rules like the chain rule. Often in calculus exercises, one function can be seen operating inside another, creating a composite function.

• For example, if your original function is \(y = (3x - 2)^6\), you can decompose it into \(u = 3x - 2\) and \(y = u^6\).
• This separation clarifies the role each part plays and makes applying calculus operations more straightforward.

Such decomposition is crucial in making sense of the function's structure, leading to accurate and easier differentiation. By reducing a function step by step, tackling each piece individually, your overall grasp of the function improves significantly.