Differentiation

Differentiation is a fundamental concept in calculus that measures how a function changes as its input changes. It helps us find the rate of change of a function.

1. Definition of the Derivative
The derivative of a function $f(x)$ with respect to $x$, denoted as $f'(x)$ or $\frac{dy}{dx}$, is defined by the limit:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

This limit represents the slope of the tangent line to the graph of $f(x)$ at any point $x$.

2. Basic Differentiation Rules

• Constant Rule: The derivative of a constant is zero. If $f(x) = c$, where $c$ is a constant, then

  $$\frac{d}{dx}(c) = 0$$

• Power Rule: The derivative of $x^n$ is $nx^{n-1}$. If $f(x) = x^n$, where $n$ is any real number, then

  $$\frac{d}{dx}(x^n) = nx^{n-1}$$

• Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function. If $f(x) = c \cdot g(x)$, where $c$ is a constant, then

  $$\frac{d}{dx}(c \cdot g(x)) = c \cdot \frac{d}{dx}(g(x))$$

• Sum/Difference Rule: The derivative of a sum or difference of functions is the sum or difference of their derivatives. If $f(x) = g(x) \pm h(x)$, then

  $$\frac{d}{dx}(g(x) \pm h(x)) = \frac{d}{dx}(g(x)) \pm \frac{d}{dx}(h(x))$$

Example: Find the derivative of $f(x) = 3x^4 - 2x^2 + 5x - 7$.