Problem 218 - Calculus Volume 1

Question

Given \(y=f(u)\) 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 y with respect to u

First, find the derivative of \( y = \tan u \) with respect to \( u \). The derivative of \( \tan u \) with respect to \( u \) is \( \sec^2 u \). Thus, we have \[\frac{dy}{du} = \sec^2 u.\]

Step 2: Differentiate u with respect to x

Next, differentiate \( u = 9x + 2 \) with respect to \( x \). The derivative of \( 9x + 2 \) with respect to \( x \) is 9. Thus, we have \[\frac{du}{dx} = 9.\]

Step 3: Apply the Chain Rule

Now, apply the chain rule in Leibniz's notation to find \( \frac{dy}{dx} \):\[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.\]Substituting the derivatives we found:\[\frac{dy}{dx} = \sec^2 u \cdot 9.\]

Step 4: Substitute u back in terms of x

Since \( u = 9x + 2 \), substitute \( u \) back into the expression for \( \frac{dy}{dx} \):\[\frac{dy}{dx} = 9 \sec^2(9x + 2).\]

Key Concepts

Leibniz's notation

In calculus, Leibniz's notation is a way of denoting derivatives that makes the process of differentiation more intuitive and easier to understand, especially when applying the chain rule. This notation involves using "d" as a symbol to signify a small change. For example, in the chain rule \[\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx},\]Leibniz's notation beautifully breaks down the process into understandable parts:

• \(\frac{dy}{du}\) represents the derivative of \(y\) with respect to \(u\), indicating how \(y\) changes as \(u\) changes.
• \(\frac{du}{dx}\) is the derivative of \(u\) with respect to \(x\), showing the rate of change of \(u\) as \(x\) varies.

Applying this notation in problems helps us connect the dots between multiple functions, making differentiation across composites a much clearer task. By effectively splitting the change into manageable parts, it simplifies the handling of derivatives, especially when dealing with complex functions.

Differentiation

Differentiation is one of the fundamental operations in calculus. It involves computing the derivative of a function, which provides the rate at which a function is changing at any given point. When we're differentiating, we're essentially finding out how one quantity changes in response to another. Consider the function \(y = \tan u\). Differentiating \(y\) with respect to \(u\) gives us \(\sec^2 u\). This tells us how \(y\) changes as \(u\) changes, specifically:

• The derivative \(\sec^2 u\) is indicative of the steepness or gradient of the tangent line to \(y = \tan u\).
• In essence, it provides a measure of how rapid the change in \(y\) is concerning changes in \(u\).

Furthermore, when we have \(u = 9x + 2\), differentiating \(u\) with respect to \(x\) results in 9, which means:

• The function \(u\) increases linearly with \(x\).
• The rate of increase is constant, represented by the coefficient 9.

This process of finding derivatives is crucial as it reveals the behavior and dynamics of functions, forming the basis for calculus problem-solving.

Calculus problem solving

Solving calculus problems often involves breaking down complex processes into bite-sized, manageable steps. This not only simplifies the process but also aids in mastering the broader concepts. When faced with a function composed of other functions, such as in our problem, using tools like the chain rule becomes essential. To tackle problems like:
Given \(y = \tan u\) and \(u = 9x + 2\), how do we find \(\frac{dy}{dx}\)?
This is where the chain rule is useful:

• Determine \(\frac{dy}{du}\), which is \(\sec^2 u\), the derivative of \(y\) with respect to \(u\).
• Find \(\frac{du}{dx}\), which is 9, the derivative of \(u\) concerning \(x\).
• Apply the chain rule: \(\frac{dy}{dx} = \frac{dy}{du}\times \frac{du}{dx} = \sec^2 u \times 9\).
• Substitute back \(u = 9x + 2\) to get the final result \(\frac{dy}{dx} = 9\sec^2(9x + 2)\).

Employing these steps allows students to not only solve the problem but also understand the underpinning calculus principles. It shows how small changes lead to bigger changes and how derivatives play a role in deciphering the relationships between variables.