Calculus - Finding the derivative of a function using limits

Introduction

This tutorial will guide you through the process of finding the derivative of a function using limits. Understanding this method is foundational in calculus and will be beneficial as you learn more advanced techniques for finding derivatives.

Step 1: Understand the Difference Quotient

To find the derivative using limits, you need to compute the difference quotient defined as:

[ \frac{f(x + h) - f(x)}{h} ]

This expression measures the average rate of change of the function over an interval as the interval approaches zero.

Step 2: Set Up the Limit

Let's find the derivative of the function ( f(x) = 5x^3 ).

1. Write the limit expression: [ \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

2. Substitute ( f(x + h) ):

   • Replace ( x ) with ( x + h ): [ f(x + h) = 5(x + h)^3 ]

3. Write the difference quotient: [ \lim_{h \to 0} \frac{5(x + h)^3 - 5x^3}{h} ]

Step 3: Expand the Function

You need to expand ( (x + h)^3 ) to simplify the expression:

1. Use the binomial expansion: [ (x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 ]

2. Substitute back into the limit expression: [ \lim_{h \to 0} \frac{5(x^3 + 3x^2h + 3xh^2 + h^3) - 5x^3}{h} ]

3. Simplifying gives: [ \lim_{h \to 0} \frac{15x^2h + 15xh^2 + 5h^3}{h} ]

Step 4: Factor and Cancel

Now, simplify by factoring out ( h ):

1. Rewrite the expression: [ \lim_{h \to 0} \frac{h(15x^2 + 15xh + 5h^2)}{h} ]

2. Cancel ( h ) from the numerator and denominator (as long as ( h \neq 0 )): [ \lim_{h \to 0} (15x^2 + 15xh + 5h^2) ]

Step 5: Evaluate the Limit

Evaluate the limit as ( h ) approaches zero:

1. Substitute ( h = 0 ): [ 15x^2 + 15(0) + 5(0)^2 = 15x^2 ]

2. Therefore, the derivative of ( f(x) = 5x^3 ) is: [ f'(x) = 15x^2 ]

Conclusion

You have successfully found the derivative of the function ( f(x) = 5x^3 ) using limits, resulting in ( f'(x) = 15x^2 ). This process highlights the importance of the difference quotient and limits in calculus. As you progress, you will learn shortcuts for finding derivatives, making the process more efficient. For further practice and more advanced topics, consider exploring additional videos and tutorials.