Meaning of differentiation

Connect the transition from average to instantaneous rate of change with tangents.

Meaning

The derivative at a point is the limit of average rates of change as the interval shrinks to that point. It represents instantaneous rate of change and can be interpreted as the graph's tangent slope.

Example

For the average rate of change between 3 and is As approaches 0, this approaches 6, so the derivative at 3 is 6.For the average rate, h must be nonzero. Cancel h first, then examine how 6+h approaches 6; do not substitute h=0 into the original quotient and divide by zero.A tangent's slope is not the function value. A function can be continuous but not differentiable at a sharp corner where the rates from the two sides differ.

Practice for this concept

Scope and references

Topics

Average and instantaneous rates of change, derivative at a point, tangent slope

Learning goal

Connect the transition from average to instantaneous rate of change with tangents.

Helpful background

Compare changes in two quantities and connect them with a function limit.

Instantaneous rates and derivatives at a point

Does not cover general differentiation formulas or tangent-line equations.

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