Question

Finding a Second Derivative In Exercises \(85-90\) , find the second derivative of the function. $$ f(x)=6\left(x^{3}+4\right)^{3} $$

Short answer

The second derivative is \(f''(x) = 108x(x^3 + 4)^2 + 648x^4(x^3 + 4)\)

Step-by-step solution

Find the first derivative

We start by applying the chain rule and the power rule. Let's denote \(u = x^3+4\). We have a function \(f(x)=6u^3\). The derivative of \(f(x)\) is calculated as: \(f'(x) = 6 *3u^2 * u'\), where \(u'=3x^2\) is the derivative of \(u=x^3+4\). Hence, \(f'(x) = 6 *3 * (x^3+4)^2 * 3x^2 = 54x^2(x^3+4)^2\).

Find the second derivative

To find the second derivative, we need to take the derivative of the first derivative. We will use the chain rule again, considering \(v = x^3+4\). We have the function \(f'(x)=54x^2v^2\). Applying the power rule first, we get \(f''(x) = 54*2x*v^2 + 54x^2 * 2v * v'\), where \(v'=3x^2\). Hence, \(f''(x) = 108x(x^3+4)^2 + 108x^2 * 2 * (x^3+4) * 3x^2 = 108x(x^3 + 4)^2 + 648x^4(x^3 + 4)\)

Key concepts

Chain Rule

Understanding the chain rule is essential for calculating derivatives, especially when dealing with composite functions. In essence, the chain rule helps you differentiate a function that is nested within another. Imagine a Russian doll setup, where each doll is a function within a function.

To apply the chain rule, you first need to identify the inner function (\( u \text{ or } v \text{ in our problem}\)) and the outer function. In the given exercise, the function \( f(x)=6(x^{3}+4)^{3} \) can be viewed as the outer function \(6u^3\) with the inner function as \(u=x^3+4\). That's our first Russian doll.

Here's the formula for the chain rule: \[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \] where \( g(x) \) is the inner function and \( f(g(x)) \) is the composite function. Using this formula, we differentiate the outer function, keeping the inner function as is, then multiply by the derivative of the inner function. This way, we unwrap the Russian dolls one by one to find the derivative of the complex, nested function.

Power Rule

The power rule is a powerful tool (pun intended) for derivatives, especially when working with expressions raised to a power. Simply put, if you have a function \( f(x) = x^n \), where \( n \) is any real number, the power rule states that the derivative of this function is \( f'(x) = n \cdot x^{n-1} \).

In our exercise, the power rule comes into play for both the first and second derivative calculations. The term \( (x^3+4)^2 \) in \( f'(x) = 54x^2(x^3+4)^2 \) is squared, and thus, when we calculate its derivative, we use the power rule to bring down the exponent of 2 in front and decrease the exponent by 1, leading us to \( 2(x^3+4) \) times the derivative of the inner function, which once more involves the chain rule.

Derivative of a Function

At the core of calculus is the concept of the derivative of a function. This fundamental process gives us the slope or rate of change at any point on a curve. The derivative can be described as the instant change in the function's output with respect to a change in its input—in other words, how fast the `y` value (output) of a function is changing at a specific `x` value (input).

For the problem at hand, finding the first and second derivatives is akin to peeling layers off an onion. The first derivative \( f'(x) \) gives us the rate at which \( f(x) \) is changing, and the second derivative, \( f''(x) \) tells us about the curvature of \( f(x) \)—how the rate of change is itself changing.

This double derivative is particularly useful for determining concavity and inflection points on a graph. For example, if \( f''(x) > 0 \) for all `x`, the graph is concave up, like a smiling face. As you dive into calculus, you'll find that understanding the first and second derivatives not only helps with graph analysis but also with many real-world applications like optimization and motion physics.