Calculus: Differentiation and Its Applications

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Calculus: Differentiation and Its Applications

What Is Differentiation?

Differentiation is a fundamental concept in calculus that deals with the rate of change of a function. The derivative represents the slope or instantaneous rate of change at a specific point.

The Derivative: Basic Definition

The derivative of a function $f(x)$ is defined as:

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

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

Basic Differentiation Rules

Power Rule

If $f(x) = x^n$ , then:

$f'(x) = n \cdot x^{n-1}$

Example:

$\text{If } f(x) = x^3, \text{ then } f'(x) = 3x^2$

Constant Rule

If $f(x) = c$ (where $c$ is constant):

$f'(x) = 0$

Sum and Difference Rule

For $f(x) = g(x) \pm h(x)$ :

$f'(x) = g'(x) \pm h'(x)$

Product Rule

For $f(x) = u(x) \cdot v(x)$ :

$f'(x) = u'(x)v(x) + u(x)v'(x)$

Quotient Rule

For $f(x) = \frac{u(x)}{v(x)}$ :

$f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}$

Applications of Differentiation

Finding Tangents to Curves

The derivative gives the slope of the tangent line at any point on a curve.

Example: Find the tangent to $y = x^2$ at $x = 3$ .

Differentiate: $y' = 2x$
Evaluate at $x = 3$ : $y' = 6$
Equation: $y - 9 = 6(x - 3)$ or $y = 6x - 9$

Velocity and Acceleration

For position $s(t)$ :

Velocity: $v(t) = \frac{ds}{dt}$
Acceleration: $a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2}$

Optimization Problems

Critical points (maxima/minima) occur where $f'(x) = 0$ .

Example Problem

Problem: Find critical points of $f(x) = 3x^4 - 4x^3 + 2x^2 - 5$ .

Solution:

Differentiate:
$f'(x) = 12x^3 - 12x^2 + 4x$
Set $f'(x) = 0$ :
$4x(3x^2 - 3x + 1) = 0$
Only real solution: $x = 0$

Common Mistakes

Misapplying the Power Rule (e.g., forgetting to subtract 1 from the exponent).
Confusing Product Rule with Sum Rule.
Incorrectly applying the Quotient Rule numerator order.

Practice Questions

Differentiate $f(x) = 5x^3 + 2x^2 - x + 7$ .
Find the tangent to $y = x^3 - 4x^2 + 2x$ at $x = 2$ .
Given $s(t) = t^2 - 4t + 5$ , find velocity at $t = 3$ .

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