Problem 6 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. \(f(x)=\sqrt{2 x+7}\)

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{1}{\sqrt{2x + 7}} \).

Step 1: Identify the Function Structure

The given function is \( f(x) = \sqrt{2x + 7} \). This is a composition of functions where the square root function is applied to another function \( 2x + 7 \).

Step 2: Use Chain Rule for Differentiation

To differentiate a composite function \( g(h(x)) \), we use the chain rule: \((g(h(x)))' = g'(h(x)) \cdot h'(x)\). For \( f(x) = \sqrt{u} \) where \( u = 2x+7 \), this becomes \( f'(x) = \frac{1}{2\sqrt{u}} \cdot (2x+7)' \).

Step 3: Differentiate the Inner Function

Differentiate \( 2x+7 \) with respect to \( x \): The derivative is \( \frac{d}{dx}(2x+7) = 2 \).

Step 4: Apply the Chain Rule

Substitute back the derivative of the inner function: \( f'(x) = \frac{1}{2\sqrt{2x + 7}} \cdot 2 \). Simplify to get \( f'(x) = \frac{1}{\sqrt{2x + 7}} \).

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus for differentiating composite functions. When you have a function nested inside another, you use the chain rule to differentiate it efficiently.
It's important to break the function down into its inner and outer components in your mind first.

• Outer function: This is the function on the outside, like the square root in \( f(x) = \sqrt{2x+7} \).
• Inner function: This is the part inside the outer function, such as \( 2x+7 \).

To apply the chain rule, differentiate the outer function with respect to the inner function, and then multiply by the derivative of the inner function. This gives you an efficient way to handle derivatives of these more complex, layered functions.

Composite Function

A composite function, or nested function, involves applying one function to the results of another. This might sound complex, but it's as simple as putting one thing inside another! In calculus, this is like saying you're plugging one formula inside another formula.Consider our example: \( f(x) = \sqrt{2x+7} \). Here, \( \sqrt{x} \) is the outer function, and \( 2x+7 \) is the inner function.

• Rather than dealing with it all at once, break it down.
• Think of the inner function first, evaluate it, and then apply the outer function.

Seeing a function in layers like this helps you visualize and differentiate it using the chain rule. It's all about recognizing the composition and dealing with each element one piece at a time.

Derivative Calculation

Derivative calculation involves finding the rate at which one quantity changes with respect to another. When dealing with a composite function, the actual differentiation is a step-by-step process.Start by understanding what's to be differentiated: In \( f(x) = \sqrt{2x+7} \), you first find the derivative of the inner function, \( 2x+7 \), which is simply \( 2 \).
Then, differentiate the outer function, replacing the inner with a 'placeholder' variable, like \( u = 2x + 7 \). The derivative of \( \sqrt{u} \) is \( \frac{1}{2\sqrt{u}} \).

• Always differentiate inside-out. Begin with the innermost function.
• Apply the chain rule accurately, multiplying the derivatives as you unfold each layer.

Finally, substitute back the original inner function and simplify. This step-by-step approach ensures you derive the composite function accurately without skipping any crucial piece of the calculation.