Problem 12 - The Calculus with Analytic Geometry

Question

Find \(D_{t}{\underline{\phantom{xx}}}^{3} s\) if \(s=\sqrt{4 t+1}\).

Step-by-Step Solution

Answer

The third derivative is \( 24 (4t + 1)^{-5/2} \).

Step 1: Identify the given function

The problem gives the function as: \[ s = \sqrt{4t + 1} \]

Step 2: Compute the first derivative

To find the first derivative of \( s \) with respect to \( t \), use the chain rule. \[ \frac{d}{dt} (\sqrt{4t + 1}) = \frac{1}{2} ( 4t + 1 )^{-1/2} \times 4 \] Which simplifies to: \[ \frac{2}{\sqrt{4t + 1}} \]

Step 3: Compute the second derivative

To find the second derivative, apply the derivative operator to the first derivative. \[ \frac{d}{dt} \frac{2}{\sqrt{4t + 1}} \] Using the chain rule again, this becomes: \[ 2 \times (4t + 1)^{-3/2} \times (-2) \] Which simplifies to: \[ -4 (4t + 1)^{-3/2} \]

Step 4: Compute the third derivative

To find the third derivative, apply the derivative operator to the second derivative. \[ \frac{d}{dt} [-4 (4t + 1)^{-3/2}] \] Using the chain rule, this becomes: \[ -4 \times (-3/2) \times (4t + 1)^{-5/2} \times 4 \] Which simplifies to: \[ 24 (4t + 1)^{-5/2} \]

Key Concepts

chain rule

The chain rule is a fundamental technique in calculus. It helps when you need to differentiate a composite function. Essentially, it states that if you have a function composed of two functions, say \(h(x) = f(g(x))\), then its derivative is formed by multiplying the derivative of the outer function by the derivative of the inner function. For a clearer understanding, let’s take \(s = \sqrt{4t+1}\). First, we set \(u = 4t + 1\). Thus, our function becomes \(s = \sqrt{u}\). The chain rule tells us:
\[ \frac{ds}{dt} = \frac{ds}{du} \times \frac{du}{dt} \]
This step-by-step approach can significantly simplify the differentiation process.

first derivative

The first derivative helps you understand how a function changes at any given point. For the function \(s = \sqrt{4t + 1}\), we apply the chain rule. First, let’s rewrite \(s\) using an exponent: \(s = (4t + 1)^{1/2}\).
Using the chain rule, the first derivative becomes:
\[ \frac{d}{dt} (4t + 1)^{1/2} = \frac{1}{2} (4t + 1)^{-1/2} \times 4 \]
Simplifying this gives:
\[ \frac{2}{\sqrt{4t + 1}} \]
This first derivative tells us the rate of change of \(s\) concerning \(t\).

second derivative

The second derivative gives insight into the concavity or curvature of a function. For \(s = \sqrt{4t + 1}\), we already found the first derivative: \( \frac{2}{\sqrt{4t + 1}} \). To find the second derivative, we again use the chain rule:
\[ \frac{d}{dt} \frac{2}{\sqrt{4t + 1}} \]
This becomes:
\[ 2 \times (4t + 1)^{-3/2} \times (-2) \]
Simplifying, we get:
\[ -4 (4t + 1)^{-3/2} \]
This second derivative indicates how the rate of change is itself changing, providing further depth in understanding the behavior of \(s\).

third derivative

The third derivative informs us about the rate of change of the second derivative, offering insights into the jerk or how the acceleration changes over time. For \(s = \sqrt{4t + 1}\), we need to differentiate the second derivative:
\[ \frac{d}{dt} [-4 (4t + 1)^{-3/2}] \]
Applying the chain rule for the third time, we get:
\[ -4 \times (-3/2) \times (4t + 1)^{-5/2} \times 4 \]
Which simplifies to:
\[ 24 (4t + 1)^{-5/2} \]
This final derivative shows the nuanced changes in the function's rate of change at an even deeper level, adding another layer to our understanding of \(s\).