Problem 7 - Calculus Single Variable

Question

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(\sqrt{6 x+5}\)

Step-by-Step Solution

Answer

The first derivative is \( \frac{3}{\sqrt{6x+5}} \), the second is \( -\frac{9}{(6x+5)^{3/2}} \), and the third is \( \frac{81}{(6x+5)^{5/2}} \).

Step 1: Rewrite the Expression

The function given is \( f(x) = \sqrt{6x + 5} \). It is helpful to rewrite the square root in terms of an exponent. Recall that \( \sqrt{a} = a^{1/2} \). Therefore, \( f(x) = (6x + 5)^{1/2} \).

Step 2: Find the First Derivative

To find the first derivative, use the chain rule. The chain rule states that if you have a composite function \( g(h(x)) \), then \( \frac{d}{dx}[g(h(x))] = g'(h(x)) \cdot h'(x) \). Here, \( g(u) = u^{1/2} \) and \( h(x) = 6x + 5 \).Taking the derivative of the outside function, \( g(u) \), we get \( \frac{1}{2}u^{-1/2} \). This leads to:\[ f'(x) = \frac{1}{2} (6x + 5)^{-1/2} \cdot 6 \]Thus:\[ f'(x) = \frac{6}{2(6x + 5)^{1/2}} = \frac{3}{\sqrt{6x+5}} \]

Step 3: Compute the Second Derivative

To find the second derivative, differentiate \( f'(x) = \frac{3}{\sqrt{6x+5}} \). Rewrite \( f'(x) \) as \( 3(6x + 5)^{-1/2} \) and apply the chain rule again:\( u = 6x + 5 \), so \( \frac{d}{dx} u^{-1/2} = -\frac{1}{2} u^{-3/2} \cdot 6 \).Differentiating gives:\[ f''(x) = 3 \cdot -\frac{1}{2} (6x+5)^{-3/2} \cdot 6 \] Simplifying, we find:\[ f''(x) = -\frac{9}{(6x+5)^{3/2}} \]

Step 4: Derive the Third Derivative

For the third derivative, differentiate \( f''(x) = -\frac{9}{(6x+5)^{3/2}} \) again:Rewrite \( f''(x) \) as \( -9(6x+5)^{-3/2} \) and apply the chain rule:\( u = 6x + 5 \), so \( \frac{d}{dx} u^{-3/2} = -\frac{3}{2} u^{-5/2} \cdot 6 \).Thus:\[ f'''(x) = -9 \cdot -\frac{3}{2} (6x+5)^{-5/2} \cdot 6 \] Simplify:\[ f'''(x) = \frac{81}{(6x+5)^{5/2}} \]

Key Concepts

Chain Rule

The chain rule is a useful technique when dealing with composite functions, where one function is nested inside another. In essence, the chain rule helps us differentiate a function that has another function inside it.
For example, if we have a function of the form \( g(h(x)) \), the chain rule teaches us to take the derivative of the outer function \( g \) with respect to the inner function \( h \), and then multiply this by the derivative of \( h(x) \) itself:

• Differentiate \( g \) with respect to \( h \).
• Multiply by the derivative of \( h(x) \).

By applying the chain rule, we can unravel even complex nested functions, making it easier than separately handling each part. This approach is tremendously helpful in various calculus problems, especially when functions are combined in this manner.

First Derivative

The first derivative of a function is essentially the rate of change or the slope of the function at any given point. When differentiating \( f(x) = (6x + 5)^{1/2} \), we apply the chain rule as discussed. This involves differentiating the outer function \((u^{1/2})\) and adjusting with the derivative of the inner function \((6x + 5)\).
This results in:

• Differentiate the outer: yields \( \frac{1}{2}u^{-1/2} \).
• Derivate the inner: yields \(6\).

Subsequently, the first derivative becomes \( f'(x) = \frac{3}{\sqrt{6x+5}}\), indicating how the function rises or falls as \(x\) varies.

Second Derivative

Finding the second derivative involves differentiating the first derivative. In this context, we analyze the second derivative to understand how the function's rate of change is itself changing over time.
For \( f'(x) = \frac{3}{\sqrt{6x+5}} \), rewrite it as \(3(6x + 5)^{-1/2} \) and apply the chain rule again:

• Differentiate using \( u^{-1/2} \).
• Account for the inside \(6x + 5\).

This process results in \( f''(x) = -\frac{9}{(6x+5)^{3/2}} \). Negative sign indicates the rate of change of the slope is decreasing; thus, reflecting on concavity (how the graph curves).

Third Derivative

Proceeding to the third derivative, we differentiate the second derivative \( f''(x) = -\frac{9}{(6x+5)^{3/2}} \). This helps in assessing the change in the function's concavity further. Applying the chain rule as before:

• Start from \( u^{-3/2} \).
• Include \(6x + 5\) effect.

The result \( f'''(x) = \frac{81}{(6x+5)^{5/2}} \) shows another layer of rate of change—how the curvature itself evolves. Contributing to insights in mechanics, economics, and various fields where higher-order changes matter.