Problem 25 - The Calculus with Analytic Geometry

Question

Given \(f(x)=x^{3}\) and \(g(x)=f\left(x^{2}\right)\). Find: (a) \(f^{\prime}\left(x^{2}\right)\); (b) \(g^{\prime}(x)\).

Step-by-Step Solution

Answer

(a) \(3x^4\); (b) \(6x^5\)

Step 1 - Find the derivative of \(f(x)\)

First, determine the derivative of the given function \(f(x) = x^3\). Using the power rule, the derivative of \(f(x)\) is given by:\[ f'(x) = \frac{d}{dx} (x^3) = 3x^2 \]

Step 2 - Evaluate \(f'(x^2)\)

Next, substitute \(x^2\) into the derivative \(f'(x)\). Therefore:\[ f'(x^2) = 3(x^2)^2 = 3x^4 \]

Step 3 - Write the composition function \(g(x)\)

The function \(g(x)\) is defined as \(g(x) = f(x^2)\). Substituting \(f(x^2)\) gives:\[ g(x) = (x^2)^3 = x^6 \]

Step 4 - Find the derivative of \(g(x)\)

Now, determine the derivative of \(g(x) = x^6\). Using the power rule, the derivative of \(g(x)\) is given by:\[ g'(x) = \frac{d}{dx} (x^6) = 6x^5 \]

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus used to find the derivative of a function of the form \(f(x) = x^n\). Here's how it works. First, differentiate \(f(x)\) by bringing the exponent \(n\) down as a coefficient. Then, subtract 1 from the original exponent. For example, if you have \(f(x) = x^3\), the derivative is \(f'(x) = 3x^2\). The power rule is a quick and simple method, making it one of the first techniques students learn when studying derivatives.
Step-by-step example:

• Function: \(f(x) = x^4\)
• Derivative: \(f'(x) = 4x^3\)

This procedure is used repeatedly, and understanding it is key to mastering other, more complex differentiation techniques.

Chain Rule

The chain rule helps you find the derivative of composite functions. When you have a function inside another function, the chain rule is what you need. Mathematically, if you have \(y = f(g(x))\), the chain rule states that the derivative \(y'\) is \(f'(g(x)) \times g'(x)\).
Let's look at how we apply the chain rule using the given functions in the exercise:

• Given \(f(x) = x^3\), we first find \(f'(x) = 3x^2\).
• For \(g(x) = f(x^2)\), substitute \(x^2\) into the derivative of \(f(x)\) to get \(f'(x^2) = 3(x^2)^2 = 3x^4\).
• Using the chain rule, find the derivative \(g'(x) = f'(x^2) \times (2x)\textrm{which is } 3x^4 \times 2x = 6x^5\).

This process will ensure you accurately differentiate composite functions.

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. If you have two functions \(f(x)\) and \(g(x)\), you can compose them to get \(g(f(x))\). This is like building a more complex function from simpler ones.
In the exercise, we have \(f(x) = x^3\) and \(g(x) = f(x^2) = (x^2)^3 = x^6\). Here, \(f(x^2)\) means you first apply \(x^2\) and then the cube function \((...)^3\). Using known derivatives and the power rule,
we break down the problem effectively.
Function composition is crucial for deeper understanding and problem-solving in calculus.