Differential calculus

Definition

Differential calculus is the branch of mathematics that deals with the study of how functions change when their inputs change. It primarily focuses on concepts such as derivatives and rates of change.

5 Must Know Facts For Your Next Test

1. The derivative of a function at a point gives the slope of the tangent line to the function's graph at that point.
2. Derivatives can be used to find rates of change, such as velocity from a position-time function.
3. The limit definition of a derivative is given by $\lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}$.
4. Differentiable functions are always continuous, but not all continuous functions are differentiable.
5. Common rules for differentiation include the power rule, product rule, quotient rule, and chain rule.

Review Questions

• What does the derivative of a function represent geometrically?
• How do you find the derivative using the limit definition?
• Can you provide an example where a function is continuous but not differentiable?

Related terms

Derivative: A measure of how a function changes as its input changes; mathematically represented as $f'(x)$ or $\frac{df}{dx}$.

Limit:

The value that a function (or sequence) approaches as the input (or index) approaches some value.

Continuity:

\text{A property of a function if it is intuitively 'smooth' or unbroken; formally, } f(x) \text{ is continuous at } x=a \text{ if } \lim_{{x \to a}} f(x) = f(a).