Problem 26 - Calculus for Biology and Medicine

Question

Use the formal definition to find the derivative of \(f(x)=\frac{1}{x+1}\) at \(x=0\).

Step-by-Step Solution

Answer

The derivative at \( x=0 \) is \( f'(0) = -1 \).

Step 1: Understand the Derivative Definition

The derivative of a function at a point, using the formal definition, is given by the limit formula: \[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]Here, we want to find \( f'(0) \). So, \( a = 0 \) and our function is \( f(x) = \frac{1}{x+1} \).

Step 2: Substitute into the Derivative Formula

Apply the function \( f(x) = \frac{1}{x+1} \) into the derivative definition:\[ f'(0) = \lim_{h \to 0} \frac{f(0+h) - f(0)}{h} \]This becomes:\[ f'(0) = \lim_{h \to 0} \frac{\frac{1}{h+1} - \frac{1}{1}}{h} \]

Step 3: Simplify the Expression

Simplify the expression inside the limit:\[ \frac{\frac{1}{h+1} - 1}{h} = \frac{\frac{1 - (h+1)}{h+1}}{h} = \frac{\frac{-h}{h+1}}{h} \]This simplifies further to:\[ \frac{-h}{h(h+1)} = \frac{-1}{h+1} \]

Step 4: Compute the Limit

Now, compute the limit as \( h \) approaches 0:\[ f'(0) = \lim_{h \to 0} \frac{-1}{h+1} = \frac{-1}{1} = -1 \]

Step 5: State the Result

The derivative of \( f(x) = \frac{1}{x+1} \) at \( x = 0 \) is \( f'(0) = -1 \).

Key Concepts

Derivative

The concept of a derivative is fundamental in calculus. It represents the rate at which a function's value changes at any given point.
Imagine you are driving, and the speedometer gives you your speed at every instance, your speed is the derivative of your travel distance with respect to time.
Mathematically, if we have a function, say \( f(x) \), the derivative of that function, denoted as \( f'(x) \) or \( \frac{df}{dx} \), tells us how \( f(x) \) changes as \( x \) changes.

• A positive derivative means the function is increasing.
• A negative derivative means the function is decreasing.
• If the derivative is zero, the function might have reached a peak or a flat point.

The derivative plays a crucial role in numerous fields like physics, engineering, and economics. It's a tool that helps describe changes and aids in optimizations and predictions.

Formal Definition of Derivative

The formal definition of a derivative is a cornerstone in the study of calculus and might seem complex at first, but it simplifies the concept of 'change.'
The derivative at a point \( a \) for a function \( f(x) \) is defined using a limit:\[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]
This expression might look daunting, but here’s what it means:

• "\( h \)" is a tiny increment added to \( x \). Think of it as a small step.
• "\( f(a + h) - f(a) \)" is the change in the function’s value from \( a \) to \( a + h \).
• Division by "\( h \)" gives the rate of change, like the slope of a line.
• The limit "\( h \rightarrow 0 \)" ensures we look at an infinitesimally small change, capturing the function's instantaneous rate of change rather than an average.

This formal definition allows us to precisely calculate the derivative at any point, helping us to understand how functions behave.

Limit Formula

In calculus, the concept of a limit is pivotal. It allows mathematicians to handle notions of infinity and infinitesimally small quantities.
The limit formula used in the derivative's definition helps formalize how a function's behavior approaches a particular value.
In simple terms, the limit \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \) tells us what happens as \( h \) becomes very close to zero.

• It finds the output when the input is infinitesimally tiny.
• This helps smooth out 'jumps' and 'gaps' in understanding the function’s behavior.
• The use of limits is what ensures calculus can handle continuous change comfortably.

The limit formula is what makes calculus a powerful tool for exploring not just static points, but dynamic changes. It's essential for crafting accurate predictions and understanding the nature of continuous environments.