Limit Definition of Derivative with Solved Example

The limit definition of a derivative is the basic concept in calculus used to find how a function changes at a specific point. It gives the instantaneous rate of change, i.e., the slope of the tangent line to the graph. It is also called differentiation from first principles.

The derivative of a function f(x) at a point x = a represents the slope of the tangent line to the curve at that point. Mathematically, the derivative is defined using limits as:

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

Where:

• f(a+h) represents the function evaluated at a small distance h away from a,
• f(a) represents the function evaluated at a,
• h is the increment that approaches 0.

This can also be represented as: f'(a) = \lim_{x \to 0} \frac{f(x) - f(a)}{x-a}

Here, the limit is defined in terms of x.

This expression calculates the slope of the secant line through two points on the curve, and by taking the limit as h approaches 0, it gives the slope of the tangent line at x = a.

Geometrical Interpretation

• Start with a secant line (two points on the curve)
• As the second point approaches the first, the secant becomes a tangent line
• The slope of this tangent = derivative

Solved Examples

Example 1: Derivative of f(x) = x²

Let’s calculate the derivative of f(x) = x² using the limit definition.

f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}

First, expand (x + h)²:

(x+h)^2 = x^2 + 2xh + h^2

Now, substitute into the limit formula:

f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}

Simplify the expression:

f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h}

Factor out h from the numerator:

f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h}

Cancel h:

f'(x) = \lim_{h \to 0} (2x + h)

Finally, as h approaches 0:

f'(x) = 2x

Thus, the derivative of f(x) = x² is f'(x) = 2x, confirming the slope of the tangent line to the curve at any point x.

Example 2: Derivative of f(x) = \frac{1}{x}

Let’s find the derivative of f(x) = \frac{1}{x} using the limit definition.

f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h}

Let's solve this step-by-step:

1. Combine the fractions in the numerator: \frac{1}{x+h} - \frac{1}{x} = \frac{x - (x+h)}{x(x+h)} = \frac{-h}{x(x+h)}
   
2. Substitute into the limit formula: f'(x) = \lim_{h \to 0} \frac{\frac{-h}{x(x+h)}}{h}
   
3. Simplify: f'(x) = \lim_{h \to 0} \frac{-1}{x(x+h)}​
   
4. Take the limit as h→0: f'(x) = \frac{-1}{x^2}

Thus, the derivative of f(x)=\frac{-1}{x^2}​.

Example 3: Derivative of f(x) = 3x² + x.

Next, we calculate the derivative of a polynomial function f(x) = 3x^2 + x.

f'(x) = \lim_{h \to 0} \frac{(3(x+h)^2 + (x+h)) - (3x^2 + x)}{h}

Let's solve this step-by-step:

1. Expand 3(x+h)² + (x+h): 3(x+h)^2 + (x+h) = 3(x^2 + 2xh + h^2) + x + h = 3x^2 + 6xh + 3h^2 + x + h
   
2. Substitute into the limit formula: f'(x) = \lim_{h \to 0} \frac{(3x^2 + 6xh + 3h^2 + x + h) - (3x^2 + x)}{h}
   
3. Simplify: f'(x) = \lim_{h \to 0} \frac{6xh + 3h^2 + h}{h}
   
4. Factor out h: f'(x) = \lim_{h \to 0} (6x + 3h + 1)
   
5. Take the limit as h→0: f'(x) = 6x + 1

Thus, the derivative of f(x) = 3x² + x is 6x + 1.

Example 4: Derivative of f(x) = \sqrt{x}

Let’s find the derivative of f(x) = \sqrt{x}​.

f'(x) = \lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt{x}}{h}

Let's solve this step-by-step:

1. Multiply the numerator and the denominator by the conjugate \sqrt{x+h} + \sqrt{x}​: f'(x) = \lim_{h \to 0} \frac{(\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})}{h(\sqrt{x+h} + \sqrt{x})}
   
2. Simplify the numerator: (\sqrt{x+h})^2 - (\sqrt{x})^2 = (x+h) - x = h
   
3. Substitute into the limit formula: f'(x) = \lim_{h \to 0} \frac{h}{h(\sqrt{x+h} + \sqrt{x})}
   
4. Simplify: f'(x) = \lim_{h \to 0} \frac{1}{\sqrt{x+h} + \sqrt{x}}
   
5. Take the limit as h→0: f'(x) = \frac{1}{2\sqrt{x}}

Thus, the derivative of f(x) = \sqrt{x} is \frac{1}{2\sqrt{x}}.

Practice Questions

Question 1: Derivative of f(x) = x³ using the limit definition.

Question 2: Derivative of f(x) = 1/x using the limit definition.

Question 3: Find the derivative of f(x) = 3x² + x using the limit definition.

Question 4: Use the limit definition to differentiate f(x) = \frac{1}{1 - x}​.

Question 5: Find the derivative of f(x) = \frac{1}{x+1} using the limit definition.

Answer Key

1. f'(x) = 3x²
2. f'(x) = -\frac{1}{x^2}
3. f'(x) = 6x + 1
4. f'(x) = \frac{1}{(1 - x)^2}
5. The derivative of f(x) = \frac{1}{x+1}​ is \frac{-1}{(x+1)^2}​.

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