Problem 218 - Calculus Volume 1

Question

For the following exercises, given \(y=f(u) \quad\) and \(u=g(x),\) find \(\frac{d y}{d x}\) by using Leibniz's notation for the chain rule: \(\frac{d y}{d x}=\frac{d y}{d u} \frac{d u}{d x}\) $$y=\tan u, u=9 x+2$$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = 9 \sec^2(9x + 2) \).

Step 1: Differentiate the outer function

To solve for the derivative \( \frac{dy}{dx} \), start by differentiating \( y = f(u) = \tan u \) with respect to \( u \). The derivative of \( \tan u \) with respect to \( u \) is \( \sec^2 u \). Thus, \( \frac{dy}{du} = \sec^2 u \).

Step 2: Differentiate the inner function

Next, differentiate \( u = g(x) = 9x + 2 \) with respect to \( x \). The derivative of \( 9x + 2 \) with respect to \( x \) is \( 9 \). Therefore, \( \frac{du}{dx} = 9 \).

Step 3: Apply the Chain Rule

Now apply the chain rule: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). Substituting the derivatives from Step 1 and Step 2, we have \( \frac{dy}{dx} = \sec^2 u \cdot 9 \).

Step 4: Substitute for \( u \)

Finally, substitute \( u = 9x + 2 \) back into the equation: \( \frac{dy}{dx} = 9 \sec^2(9x + 2) \).

Key Concepts

Leibniz's Notation

Leibniz's Notation is a way of writing derivatives that makes the process of solving calculus problems more intuitive. This approach uses the notation \( \frac{dy}{dx} \) to express the derivative of \( y \) with respect to \( x \). It visually represents the derivative as a fraction, which can aid in understanding complex differentiation processes.

In the context of the Chain Rule, Leibniz's Notation is especially helpful. The chain rule states that if you have a composite function, you can find its derivative by multiplying the derivative of the outer function by the derivative of the inner function. In Leibniz's terms, this is written as:

• \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)

This notation breaks down the differentiation process into simpler steps: first finding the derivative of \( y \) with respect to an intermediate variable \( u \), and then finding the derivative of \( u \) with respect to \( x \). This step-by-step approach using Leibniz's Notation makes complex derivatives manageable and understandable.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative indicates how a function changes at any given point and is a measure of a function's rate of change. Differentiation is crucial in understanding the behavior of functions and solving real-world problems related to change.

To differentiate a function like \( y = f(u) = \tan u \) with respect to \( u \), you need to determine how the tangent function changes as \( u \) changes. The derivative of \( \tan u \) is \( \sec^2 u \), which was calculated in the solution as \( \frac{dy}{du} = \sec^2 u \).

For the function \( u = g(x) = 9x + 2 \), differentiating with respect to \( x \) yields a constant rate of change, \( 9 \), thus \( \frac{du}{dx} = 9 \). Differentiation involves these straightforward calculations that, when combined using the chain rule, solve the problem of determining the rate at which \( y \) changes with \( x \).

Composite Functions

A composite function is a function that applies one function to the results of another. In simpler terms, it's a function within a function. For example, in \( y = \tan(u) \) and \( u = 9x + 2 \), \( u \) acts as the inner function, while \( \tan(u) \) is the outer function.

Understanding composite functions is essential for applying the chain rule effectively. The chain rule allows us to differentiate composite functions systematically. It does so by differentiating each part separately—first the outer function, then the inner function—and then combining these derivatives.

Let's break it down: Finish by substituting the expression for \( u \) back in to complete the process. If \( y = \tan(9x + 2) \), you find \( \frac{dy}{dx} \) by applying the determined derivatives with the chain rule:

• Outer function derivative: \( \frac{dy}{du} = \sec^2(u) \)
• Inner function derivative: \( \frac{du}{dx} = 9 \)
• Overall derivative: \( \frac{dy}{dx} = 9 \sec^2(9x + 2) \)

Understanding how to separate and differentiate these components is crucial for simplifying and solving calculus problems involving composite functions.