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Limit Definition of Derivative with Solved Example - GeeksforGeeks
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Limit Definition of Derivative with Solved Example

Last Updated : 29 Sep, 2024
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Limit definition of a derivative is the foundational concept in calculus for understanding how functions change at a specific point. It represents the instantaneous rate of change of a function, which geometrically corresponds to the slope of the tangent line at a given point on the function's graph.

It is also called differentiation from first principles and provides a formal way to compute derivatives and helps us understand the behavior of functions near any point. In this article, we will discuss Limit Definition of Derivative including solved examples as well.

Table of Content

What is Derivatives?

In calculus, a derivative represents the rate at which a function changes as its input changes. In simple terms, it tells us how a function's output value (typically represented as y or f(x)) changes with respect to changes in the input value (usually x).

The derivative of a function at a specific point provides the slope of the tangent line to the function's graph at that point.

Limit Definition of Derivative

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}

In this formula:

  • 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)}{h}

Here 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

The derivative gives the slope of the tangent line to the curve y = f(x) at any point x. When using the limit definition, we are essentially "zooming in" on the curve so closely that the secant line (which intersects the curve at two points) becomes the tangent line (which touches the curve at just one point).

Solved Examples

Example 1: Derivative of f(x) = x2

Solution:

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}

Solution:

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}

Thuf'(x) = \lim_{h \to 0} (6x + 3h + 1)s, the derivative of f(x) = \frac{1}{x}​ is \frac{-1}{x^2}​.

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

Solution:

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}

Solution:

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

f'(x) = 3x2

f'(x) = -\frac{1}{x^2}

f'(x) = 6x + 1

f'(x) = \frac{1}{(1 - x)^2}

The derivative of f(x) = \frac{1}{x+1}​ is \frac{-1}{(x+1)^2}​.

Conclusion

In conclusion, the limit definition of a derivative serves as a fundamental building block in calculus, providing a precise way to measure how a function changes at any specific point. By using limits, we can determine the instantaneous rate of change, which translates to the slope of the tangent line at a given point on a curve.

Read More,

FAQs on Limit Definition of Derivative

What is the limit definition of a derivative differentiable?

If the limit exists for every c ∈ (a, b) then we say that f is differentiable on (a, b). Graphically, this definition says that the derivative of f at c is the slope of the. tangent line to y = f(x) at c, which is the limit as h → 0 of the slopes of the lines.

What is h in the limit definition of a derivative?

h is an arbitrary distance away from x. the idea of the derivative is you have a function f(x). the average rate of change is given by. (f(x+h) - f(x))/(x +h -x).

What is the formula for the first derivative?

 The first derivative of at is given by f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h where the limit as h approaches zero is key.


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