Basics of Calculus

The language of change: Rate (Differentiation) and Accumulation (Integration).

1. Differentiation (Rate of Change)

Physical Meaning: Slope

Differentiation finds the instantaneous rate of change of one variable with respect to another. Geometrically, $\frac{dy}{dx}$ is the Slope of the curve $y=f(x)$.

A. The Power Rule (Most Important):

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

If $y = x^3$, then $\frac{dy}{dx} = 3x^{3-1} = 3x^2$

B. Common Derivatives:

Function ($y$)	Derivative ($\frac{dy}{dx}$)
Constant ($k$)	0
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\ln x$	$1/x$
$e^x$	$e^x$

C. Chain Rule (Function inside Function):

Differentiate outer function, then multiply by derivative of inner function.

$y = \sin(2x)$
$\frac{dy}{dx} = \cos(2x) \times \frac{d}{dx}(2x)$
$= 2 \cos(2x)$

2. Integration (Accumulation)

Physical Meaning: Area

Integration is the reverse process of differentiation. Geometrically, $\int y \, dx$ represents the Area under the curve.

A. The Power Rule for Integration:

$$ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) $$

$\int x^2 \, dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3} + C$

B. Special Integral (Log):

$$ \int \frac{1}{x} \, dx = \ln|x| + C $$

C. Definite Integral (With Limits): Calculates value between $x=a$ and $x=b$. No constant ($C$) needed.

$$ \int_{a}^{b} f(x) \, dx = [F(x)]_a^b = F(b) - F(a) $$

$\int_{1}^{3} 2x \, dx = [x^2]_1^3 = (3^2) - (1^2) = 9 - 1 = 8$

3. Applications in Science

In Physics

Velocity: $v = \frac{dx}{dt}$ (Slope of x-t graph)
Acceleration: $a = \frac{dv}{dt}$ (Slope of v-t graph)
Displacement: $x = \int v \, dt$ (Area under v-t graph)
Work Done: $W = \int F \, dx$ (Area under F-x graph)

In Chemistry

Rate of Reaction: $-\frac{d[R]}{dt}$ (Slope of conc-time graph)
Integrated Rate Laws: Deriving equations like $\ln[A] = -kt + \ln[A]_0$ from differential rate laws.
Thermodynamics: $w = -\int P \, dV$ (Area under PV graph)

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