The derivative of a function at a point measures its instantaneous rate of change - the slope of the tangent line at that point. It is defined as a limit: . This expression is the slope of a secant line (connecting two points on the curve) as the two points get infinitely close together.
When the limit exists, the function is differentiable at that point and the derivative gives the exact slope of the curve at that instant.
Derivatives have countless real-world interpretations: velocity is the derivative of position with respect to time; acceleration is the derivative of velocity; marginal cost is the derivative of total cost. The power rule (), product rule, quotient rule, and chain rule form a toolkit for computing derivatives algebraically without going back to the limit definition every time. A positive derivative means the function is increasing; a negative derivative means decreasing; a zero derivative indicates a potential local maximum, minimum, or inflection point.
Understanding Derivative as Slope
A useful way to think about a derivative is as a local prediction tool. Near one chosen input, a smooth curve behaves almost like a straight line. The derivative tells how steep that nearby line is.
If the input changes by a small amount, the output changes by roughly the derivative times that small amount. This is called linear approximation.
For a function where the output is height and the input is horizontal distance, a derivative of three means the height rises about three units for each one unit move to the right near that location. The estimate becomes more reliable as the input change becomes smaller.
Units give derivatives much of their meaning. A distance measured in metres as a function of time in seconds has a derivative measured in metres per second. A temperature measured in degrees as a function of time has a derivative in degrees per minute or degrees per hour.
In science classes, students should always state these units. They show what is changing and how quickly it changes.
A steep graph does not automatically mean a large physical rate, because the scales on the axes matter. A graph can look steep simply because one axis has been stretched.
Not every point on a graph has a derivative. At a sharp corner, the slopes approached from the left and from the right can disagree. The point of a V shaped graph is a familiar example.
A vertical tangent creates another problem because its slope is not a finite number. Breaks, jumps, and holes prevent a derivative at that input as well. A function can be continuous yet fail to be differentiable at a corner.
However, a function that is not continuous cannot have a derivative there. This distinction matters when reading graphs and when deciding where a formula-based answer applies.
The derivative graph gives a second view of the original function. Where the original graph rises, its derivative graph lies above zero. Where the original graph falls, the derivative graph lies below zero.
A high point or low point on the original often matches an input where the derivative is zero, though it must be checked rather than assumed. The size of the derivative measures steepness, while changes in the derivative show whether the curve is bending upward or downward.
When solving problems, keep the original function, its first derivative, and the meaning of the variables separate. Common errors come from mixing up an output value with a rate, forgetting the chain rule for a nested expression, or treating every zero derivative as a maximum or minimum.
Key Facts
- - the limit definition of the derivative
- Power rule:
- Product rule:
- Chain rule:
- : function increasing; : decreasing; : critical point
- Second derivative test: local min; local max at a critical point
Vocabulary
- Derivative
- The instantaneous rate of change of a function at a point; geometrically, the slope of the tangent line to the graph at that point.
- Tangent line
- A line that touches a curve at exactly one point (locally) and has the same slope as the curve at that point.
- Secant line
- A line connecting two points on a curve; as the points approach each other, the secant slope approaches the derivative.
- Critical point
- A point where or is undefined; candidates for local maxima, minima, or inflection points.
- Differentiable
- A function is differentiable at a point if the derivative exists there; requires the function to be continuous and smooth (no sharp corners or cusps).
Common Mistakes to Avoid
- Confusing the average rate of change with the instantaneous rate. Average rate = over an interval; instantaneous rate = at a single point via a limit.
- Applying the power rule to exponential functions. (not ). The power rule applies when is the base and the exponent is a constant, not when is the exponent.
- Forgetting to apply the chain rule for composite functions. , not just . The derivative of the outer function must be multiplied by the derivative of the inner function.
- Concluding that guarantees a maximum or minimum. at inflection points too (e.g., at ). Use the second derivative test or sign chart to classify.
Practice Questions
- 1 Find the derivative of . At what -values is the tangent line horizontal?
- 2 Using the limit definition, find for .
- 3 A particle's position is . Find the velocity and acceleration functions and determine when the particle is at rest.