Problem 12 - Calculus an Applied Approach

Question

find the second derivative of the function. $$ y=4\left(x^{2}+5 x\right)^{3} $$

Step-by-Step Solution

Answer

The second derivative of the function \(y = 4(x^2 + 5x)^3\) is \(y'' = 48x^3 + 360x^2 + 600x\).

Step 1: Apply Chain Rule For First Differentiation

The derivative of \(y = 4(x^2 + 5x)^3\) using the chain rule (where \(u = x^2 + 5x\)) is: \[y' = 4' * (u)^3 + 4 * 3 * (u)^2 * u' = 12u^2(u') = 12(x^2+5x)^2(2x+5)\]

Step 2: Simplify the Result

Simplify the result from step 1. The simplified form of the derivative is: \[y' = 24x^2(x^2 + 5x) + 60x(x^2 + 5x)\]

Step 3: Apply Product and Chain Rule For Second Differentiation

Now we find the second derivative of the function; that is, \((y')' = y''\). Using the product rule and chain rule again, we obtain: \[y'' = 24[2x(x^2 + 5x) + x^2(2x+5)] + 60[(x^2 + 5x) + x(2x+5)]\]

Step 4: Simplify the Second Derivative

Finally, simplify the second derivative to get the final result: \[y'' = 48x^3 + 240x^2 + 120x^2 + 600x = 48x^3 + 360x^2 + 600x\]

Key Concepts

Chain Rule

The Chain Rule is an essential tool in calculus for differentiating composite functions. A composite function means that there's a function within another function, like peeling an onion. In our exercise, we have the function \( y = 4(x^2 + 5x)^3 \). To differentiate this, we first think of \( x^2 + 5x \) as a single unit, which we can call \( u \). Thus, the function becomes \( 4u^3 \). The chain rule tells us to differentiate 'layer by layer': first differentiate the outer function, then the inner part.

• First, differentiate the outer function \( 4u^3 \) with respect to \( u \), giving us \( 12u^2 \).
• Next, differentiate \( u = x^2 + 5x \) with respect to \( x \), which is \( 2x + 5 \).
• Multiply these two results together: \( 12u^2(2x+5) \) or \( 12(x^2+5x)^2(2x+5) \).

This layered approach can initially seem daunting, but once you understand the sequence it becomes much clearer. Always remember to differentiate from the outside inwards.

Product Rule

The Product Rule is used when you have a product of two functions that you need to differentiate. In this case, when finding the second derivative \( y'' \), the expression is a bit more complex. We need the product rule because we're working with multiple parts.

• Suppose you're differentiating \( f(x)g(x) \). The product rule tells us: \( (fg)' = f'g + fg' \). That means you differentiate one function while keeping the other as it is, and then switch roles.
• For our second differentiation, we apply to parts such as \( 24x^2(x^2 + 5x) \) with the simplified result.
• This requires careful application alongside the chain rule.

Combining the chain rule for the inner function and the product rule is tricky, so take it step by step. Consistent practice helps to build familiarity with when and how to use these rules together.

Differentiation

Differentiation is the process of finding the derivative, or the rate of change, of a function. This is a foundational concept in calculus used to understand how a function behaves at any given point.

• The first derivative \( y' \), calculated using the chain rule, reveals the slope of the tangent line to the curve \( y = 4(x^2 + 5x)^3 \).
• To find \( y'' \), which is the second derivative, you're determining the rate at which the first derivative changes, used often to determine the concavity or acceleration of the function.
• In this exercise, after differentiating twice, you get \( y'' = 48x^3 + 360x^2 + 600x \) which provides insights on the curvature properties of \( y \).

Remember, differentiation allows you to peel back the layers of a function to understand its underlying dynamics. Mastery of basics like chain and product rules unlocks the ability to tackle complex functions with ease.