Question

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

Short answer

The second derivative of the function \(y = 4(x^2 + 5x)^3\) is \(y'' = 72x^4 + 480x^3 + 1080x^2\).

Step-by-step solution

Find the first derivative

The first step is to find the first derivative (\(y'\)) of the function \(y = 4(x^2 + 5x)^3\). According to the chain rule of differentiation, the derivative of a composition of functions is the derivative of the outer function multiplied by the derivative of the inner function. So, the first derivative is: \(y' = 12(x^2 + 5x)^2 \cdot (2x + 5)\).

Simplify the first derivative

Next, the first derivative is simplified by expanding, so we have: \(y' = 24x(x^2 + 5x)^2 + 60(x^2 + 5x)^2\). It can be further simplified to: \(y' = 24x^3(x^2 + 5x) + 60x^2(x^2 + 5x)\).

Find the second derivative

Finally, we differentiate \(y'\) to find the second derivative (\(y''\)). Following the product and chain rule, we obtain: \(y'' = 72x^2(x^2 + 5x) + 48x^3(2x+5) + 120x(x^2 + 5x) + 120x^2(2x+5)\). There is an x factor common in every term, so the final simplified form of the second derivative is: \(y'' = 72x^4 + 360x^3 + 288x^3+480x^2 + 120x^3+600x^2\).

Key concepts

First Derivative

The first derivative of a function provides information about the rate of change of the function's value with respect to changes in its input. It's essentially how steep the function is at any given point. To find the first derivative of the function given, which is in the form of \[y = 4(x^2 + 5x)^3\]we utilize two crucial rules in calculus: the chain rule and the product rule. In this case, we use the chain rule, since it involves a composition of functions. The first derivative allows us to determine critical features of the function like whether it is increasing or decreasing at certain x-values.

In practice, finding the first derivative involves:

• Identifying the outer and inner functions in the composition.
• Applying the derivative of the outer function, while keeping the inner function unchanged.
• Multiplying the result by the derivative of the inner function.

Chain Rule

The chain rule is an essential tool in calculus used when differentiating composite functions. If you have a function within another function, the chain rule breaks this down into manageable parts. When using the chain rule, you differentiate the outer function and multiply it by the derivative of the inner function.

In our example, we had the function \[y = 4(x^2 + 5x)^3\]Here \(x^2 + 5x \)is considered the inner function, and the whole expression raised to the power of 3 is the outer function. To apply the chain rule, we:

• Differentiate the outer function: The derivative of \((something)^3 \)is \(3(something)^2 \).
• Keep the inner function as it is in the first step \((x^2 + 5x)^2 \)
• Then multiply by the derivative of the inner function, which is \(2x + 5\).

Product Rule

The product rule is another differentiation technique employed when dealing with the product of two functions. It states that if you have two functions multiplied together, the derivative is not just the product of the derivatives. Instead, it's a sum which involves both functions and their derivatives.

In formula terms, if \(u(x)\) and \(v(x)\) are functions, then the derivative is \((uv)' = u'v + uv'\).This rule is pivotal when derivating our solved function using the first derivative to find the second derivative.
Use the product rule when:

• Each term in the expression needs separate differentiation.
• It involves complicated expressions that are easier to manage as a product of two factors.

Differentiation

Differentiation is a fundamental mathematical operation in calculus that involves finding the derivative of a function. It measures how a function changes as its input changes and is vital in understanding behaviors of mathematical models.

The process is based on limits and provides a systematic approach in finding a function's instantaneous rate of change. Differentiation is used:

• To determine the slope of the function at any specific point.
• To find out points of maxima, minima, and inflections for curve sketching.
• To solve complex application problems in physics, engineering, and economics.

When you differentiate, especially beyond the first derivative, such as finding a second derivative, you are essentially observing how the derivative itself changes, giving insight into the function's concavity and acceleration dynamics.