Derivative Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

hard
Use the limit definition to find the derivative of f(x)=x2f(x) = x^2.

Solution

  1. 1
    The limit definition: fโ€ฒ(x)=limโกhโ†’0f(x+h)โˆ’f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
  2. 2
    Compute f(x+h)=(x+h)2=x2+2xh+h2f(x+h) = (x+h)^2 = x^2 + 2xh + h^2.
  3. 3
    Substitute: (x2+2xh+h2)โˆ’x2h=2xh+h2h\frac{(x^2 + 2xh + h^2) - x^2}{h} = \frac{2xh + h^2}{h}.
  4. 4
    Simplify: h(2x+h)h=2x+h\frac{h(2x + h)}{h} = 2x + h for hโ‰ 0h \neq 0.
  5. 5
    Take the limit: limโกhโ†’0(2x+h)=2x\lim_{h \to 0} (2x + h) = 2x.

Answer

fโ€ฒ(x)=2xf'(x) = 2x
The limit definition is the foundation of all derivative rules. It shows that the derivative is the limit of the difference quotient โ€” the slope of the tangent line at any point.

About Derivative

The instantaneous rate of change of a function at a single point, defined as the limit of the slope of secant lines.

Learn more about Derivative โ†’

More Derivative Examples

Example 1 easy

Find the derivative of [formula].

Example 2 medium

Find the derivative of [formula] and evaluate [formula].

Example 4 easy

Find the derivative of [formula].

Example 5 hard

Find the derivative of [formula] and evaluate [formula].

โ† All Derivative Examples Derivative Concept