Derivative

Definition

The derivative of a function at a point is the rate at which the function's value changes as its input changes. It is defined as the limit of the difference quotient as the interval approaches zero.

5 Must Know Facts For Your Next Test

1. The derivative is often denoted by $f'(x)$ or $\frac{df}{dx}$.
2. The derivative of a constant function is zero.
3. The Power Rule states that if $f(x)=x^n$, then $f'(x)=nx^{n-1}$.
4. A function is differentiable at a point if it has a defined derivative at that point.
5. The derivative can be used to find the slope of the tangent line to the curve of a function at any given point.

Review Questions

• What is the geometric interpretation of a derivative?
• How do you find the derivative using the limit definition?
• What does it mean for a function to be differentiable?

Related terms

Difference Quotient: The expression $\frac{f(x+h)-f(x)}{h}$, which approximates the slope of the secant line between two points on a graph.

Power Rule: A basic rule for differentiation stating that if $f(x)=x^n$, then $f'(x)=nx^{n-1}$.

Tangent Line: A straight line that touches a curve at exactly one point and has the same slope as the curve at that point.