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:

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:

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.

First, expand (x + h)²:

Now, substitute into the limit formula:

Simplify the expression:

Factor out h from the numerator:

Cancel h:

Finally, as h approaches 0:

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

Let’s find the derivative of using the limit definition.

Let's solve this step-by-step:

1. Combine the fractions in the numerator:
   
2. Substitute into the limit formula:
   
3. Simplify:
   
4. Take the limit as h→0:

Thus, the derivative of ​.

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

Next, we calculate the derivative of a polynomial function .

Let's solve this step-by-step:

1. Expand 3(x+h)² + (x+h):
   
2. Substitute into the limit formula:
   
3. Simplify:
   
4. Factor out h:
   
5. Take the limit as h→0:

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

Example 4: Derivative of

Let’s find the derivative of ​.

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 ​:
   
2. Simplify the numerator:
   
3. Substitute into the limit formula:
   
4. Simplify:
   
5. Take the limit as h→0:

Thus, the derivative of is .

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 ​.

Question 5: Find the derivative of using the limit definition.

Answer Key

1. f'(x) = 3x²
2. The derivative of ​ is ​.

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• Limit at Infinity