Basics of Differentiation

What is Differentiation?

Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function is changing at any given point. It is used to determine the gradient (slope) of a curve at a particular point.

In simpler terms, differentiation helps us understand how a function behaves as its input changes.

Key Concepts

• Derivative: The derivative of a function represents its rate of change. It is denoted by $f'(x)$ or $\frac{dy}{dx}$.
• Tangent Line: A line that touches a curve at a point without crossing it. The slope of this line is the derivative at that point.
• Notation: Common notations for derivatives include $f'(x)$, $\frac{dy}{dx}$, and $Df(x)$.

Basic Rules of Differentiation

• Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
• Constant Rule: The derivative of a constant is zero. If $f(x) = c$, then $f'(x) = 0$.
• Sum Rule: The derivative of a sum is the sum of the derivatives. If $f(x) = u(x) + v(x)$, then $f'(x) = u'(x) + v'(x)$.
• Difference Rule: The derivative of a difference is the difference of the derivatives. If $f(x) = u(x) - v(x)$, then $f'(x) = u'(x) - v'(x)$.

Examples

Example 1: Differentiate $f(x) = 3x^2 + 5x - 4$

Example 2: Differentiate $g(x) = 7$