Problem 22 - Calculus for Biology and Medicine

Question

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ h(x)=\sqrt[3]{1+2 x} $$

Step-by-Step Solution

Answer

The derivative of \(h(x) = \sqrt[3]{1+2x}\) is \(h'(x) = \frac{2}{3}(1+2x)^{-2/3}\).

Step 1: Recognize the Problem

We need to differentiate a cube root function with respect to the independent variable, which in this case is \(x\). Our function is \(h(x) = \sqrt[3]{1+2x}\).

Step 2: Rewrite for Differentiation

To differentiate more easily, rewrite the cube root as a power: \(h(x) = (1 + 2x)^{1/3}\). This will allow us to apply the chain rule more effectively.

Step 3: Apply the Chain Rule and Power Rule

Use the chain rule to differentiate \((1+2x)^{1/3}\). The outer function is \(u(x) = x^{1/3}\) and the inner function is \(v(x) = 1+2x\). First, differentiate the outer function: \(u'(x) = \frac{1}{3}x^{-2/3}\). Now differentiate the inner function: \(v'(x) = 2\).

Step 4: Differentiate the Composite Function

Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us:\[h'(x) = u'(v(x)) \cdot v'(x) = \frac{1}{3}(1+2x)^{-2/3} \cdot 2\]Simplify this expression to get the final derivative.

Step 5: Simplify the Expression

Carry out the multiplication to find:\[h'(x) = \frac{2}{3}(1+2x)^{-2/3}\]This is the simplified derivative of the function with respect to \(x\).

Key Concepts

Chain Rule

Differentiation of functions often requires more than just basic rules. This is where the concept of the **chain rule** comes in handy. The chain rule is a method used to differentiate composite functions. In essence, a composite function is one that is made by combining two or more functions. Let's say we have a function that can be expressed as: \( f(g(x)) \). To apply the chain rule, we need to:

• Identify the outer function and the inner function.
• Differentiate the outer function while keeping the inner function intact.
• Multiply the result by the derivative of the inner function.

In our example, differentiating \((1+2x)^{1/3}\) involves taking the derivative of the cube root (the outer function) and then multiplying it by the derivative of \(1+2x\) (the inner function). The chain rule is pivotal in breaking down complex differentiation processes into manageable steps.

Power Rule

The **power rule** is one of the most straightforward techniques in differentiation. It states that if you have a function \( x^n \), then its derivative is \( nx^{n-1} \). This rule is directly applied when you see exponential expressions.

In the problem we have:

• First, the cube root, \((1+2x)^{1/3}\), is rewritten as \((1+2x)\) raised to the power of \(1/3\).
• Applying the power rule to \(x^{1/3}\), we differentiate to get \((1/3)x^{-2/3}\).

The power rule simplifies the process of finding the derivative, especially in combination with functions that involve powers or roots. It is particularly useful in conjunction with the chain rule, as seen here.

Composite Function

A **composite function** is essentially a function composed of two or more functions. Symbolically, if \( g(x) \) and \( f(x) \) are functions, the composite function is expressed as \( f(g(x)) \).

In the problem, the function \( h(x) = \sqrt[3]{1+2x} \) is a composite function where the cube root is applied to \(1+2x\). Breaking it down:

• The inner function is \( v(x) = 1 + 2x \).
• The outer function is \( u(x) = x^{1/3} \).

To differentiate a composite function, like in our example, you use the chain rule. This rule focuses on handling the inner and outer functions separately and combining their derivatives. Recognizing and working with composite functions are essential skills, especially in calculus, as they frequently appear in real-life applications and more complex differential calculus problems.