Problem 9 - Calculus an Applied Approach

Question

Find \(d y / d u, d u / d x,\) and \(d y / d x.\) $$ y=u^{2}, u=4 x+7 $$

Step-by-Step Solution

Answer

So, the derivative \(dy/du\) is \(2u\), the derivative \(du/dx\) is \(4\), and the derivative \(dy/dx\) is \(8u\).

Step 1: Finding \(dy/du\)

Differentiate the equation \(y = u^2\) with respect to \(u\). The power rule gives us: \(dy/du = 2u\). So, the derivative of \(y\) with respect to \(u\) is \(2u\).

Step 2: Finding \(du/dx\)

Now, differentiate the equation \(u = 4x + 7\) with respect to \(x\). The derivative of a constant is 0 and of \(x\) is 1, giving us: \(du/dx = 4\). So, the derivative of \(u\) with respect to \(x\) is \(4\).

Step 3: Finding \(dy/dx\)

Now, use the chain rule \(dy/dx = dy/du * du/dx\). Plug in the given numbers from the previous steps, giving us \(dy/dx = 2u * 4\). Simplify to get: \(dy/dx = 8u\).

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes. It's a fundamental tool in calculus used for analyzing the way functions change. In simple terms, when we differentiate a function, we're looking for the slope of the function at any given point.

For instance, when we are given a position-time graph of an object, the derivative of the position function gives us the velocity of that object. Differentiation helps us understand instantaneous rates of change and is essential in fields like physics, engineering, and economics.

Power Rule

The power rule is one of the most basic and widely used rules for differentiation. It applies to functions where the variable is raised to a power and states that if you have a function of the form \( f(x) = x^n \), where \( n \) is any real number, the derivative of \( f \) with respect to \( x \) is \( f'(x) = nx^{n-1} \).

Applying the power rule makes finding derivatives straightforward for polynomial functions. For example, in the given exercise, to find \( dy/du \) for \( y=u^2 \), we simply multiply the exponent by the base and subtract one from the exponent, resulting in \( 2u^{2-1} \), which simplifies to \( 2u \).

Derivative Calculation

Derivative calculation involves applying differentiation rules to compute the derivative of a given function. In the provided exercise, we calculated the derivative of \( u \) with respect to \( x \) by differentiating \( u=4x+7 \). The derivative of a constant is zero, and the derivative of \( x \) with respect to \( x \) is one. Thus, the derivative of \( 4x \) is \( 4 \) and the derivative of \( 7 \) is zero, therefore \( du/dx = 4 \).

By combining these simple differentiation results using the rules for constants and the power rule, one can easily find the derivatives for more complex functions.

Composite Function Differentiation

Composite function differentiation is used when you have a function inside another function, commonly referred to as a 'function of a function.' The chain rule is the tool we use in calculus to differentiate composite functions. It states that if you have two functions \( u(x) \) and \( y(u) \), then the derivative of \( y \) with respect to \( x \) is the product of the derivative of \( y \) with respect to \( u \) and the derivative of \( u \) with respect to \( x \). In notation, the chain rule is represented as \( dy/dx = (dy/du) * (du/dx) \).

In our exercise, after finding \( dy/du = 2u \) and \( du/dx = 4 \), we use the chain rule to find \( dy/dx = 2u * 4 \). This final step links the rates of change of the functions, and with substitution of \( u \) with its expression in terms of \( x \), we can calculate the derivative of the entire composite function.