Calculus: Understanding Derivatives

Derivative Definition

• The derivative of a function $f$ at a number $a$, denoted by $f'(a)$, is defined as $f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$ if this limit exists.
• An equivalent definition: $f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$

Derivative Interpretations

• The derivative represents the slope of the tangent line to $y=f(x)$ at $x=a$.
• The equation of the tangent line is given by: $y-f(a) = f'(a)(x-a)$
• The derivative also represents the instantaneous rate of change of $y=f(x)$ with respect to $x$ at $x=a$.

Derivative Notation

• If $y = f(x)$, the derivative can be represented as: $f'(x) = y' = \frac{dy}{dx} = \frac{df}{dx} = \frac{d}{dx}f(x) = Df(x) = D_xf(x)$

Derivatives of Basic Functions

• The derivative of a constant $c$ is 0: $\frac{d}{dx}(c) = 0$
• The derivative of $x^n$ is $nx^{n-1}$: $\frac{d}{dx}(x^n) = nx^{n-1}$
• The derivative of $e^x$ is $e^x$: $\frac{d}{dx}(e^x) = e^x$
• The derivative of $a^x$ is $a^x\ln(a)$: $\frac{d}{dx}(a^x) = a^x\ln(a)$

Differentiation Rules

• Constant Multiple Rule: $\frac{d}{dx}[cf(x)] = c\frac{d}{dx}f(x)$
• Sum/Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}f(x) \pm \frac{d}{dx}g(x)$
• Product Rule: $\frac{d}{dx}[f(x)g(x)] = f(x)\frac{d}{dx}g(x) + g(x)\frac{d}{dx}f(x)$
• Quotient Rule: $\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{[g(x)]^2}$

Chain Rule

• The chain rule is expressed as: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
• Or equivalently: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$