Differentiation

Definition

Differentiation is the process of finding the derivative of a function, which measures how a function's output changes as its input changes. It provides the slope of the tangent line to the function at any given point.

5 Must Know Facts For Your Next Test

1. The derivative of a function $f(x)$ at a point $x$ is denoted as $f'(x)$ or $\frac{df}{dx}$.
2. The power rule for differentiation states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
3. The derivative represents the instantaneous rate of change of the function with respect to its variable.
4. Common differentiation rules include the product rule, quotient rule, and chain rule.
5. If a function is differentiable at a point, it must also be continuous at that point.

Review Questions

• What is the derivative of $f(x) = x^3$ using the power rule?
• Explain how to find the derivative of a product of two functions.
• Why must a differentiable function be continuous?

Related terms

Derivative: A measure of how a function's output value changes as its input value changes. Denoted by $f'(x)$ or $\frac{dy}{dx}$.

Product Rule: A formula used to find the derivatives of products of two functions: $(fg)' = f'g + fg'$.

Chain Rule: A formula for computing the derivative of the composition of two or more functions: $(f(g(x)))' = f'(g(x))g'(x)$.