Engneering Mathematics

2. Differential Calculus

1. Derivatives and Their Physical Meaning

A derivative represents the rate at which a function changes with respect to a variable. Mathematically, if

y = f(x)

$y = f (x)$ , the derivative

f'(x)

f′(x) or

\frac{dy}{dx}

dxdy measures how

y

y changes as

x

x changes.

Physical Interpretations:

Slope of a Curve: The derivative gives the slope of the tangent line at any point on a function’s graph.
Velocity in Physics: If $s(t)$ s(t) represents position at time $t$ t, then $v(t) = s'(t)$ $v (t) = s' (t)$ is the velocity (rate of change of position).
Rate of Growth: In economics and biology, derivatives measure growth rates, such as population increase or profit maximization.

Example:
If

f(x) = x^2

$f (x) = x 2$ , then the derivative is

f'(x) = 2x

$f' (x) = 2 x$ . This means at

x = 3

$x = 3$ , the slope of the curve is

2(3) = 6

$2 (3) = 6$ .

2. Rules of Differentiation

To compute derivatives efficiently, calculus provides several differentiation rules:

a) Power Rule

f(x) = x^n

$f (x) = x n$ , then

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

$f' (x) = n x n - 1$

Example:

f(x) = x^3 \Rightarrow f'(x) = 3x^2

$f (x) = x 3 \Rightarrow f' (x) = 3 x 2$

b) Product Rule

f(x) = u(x) \cdot v(x)

$f (x) = u (x) \cdot v (x)$ , then

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

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

Example:

f(x) = x^2 \sin x

$f (x) = x 2 sin x$ , then

f'(x) = 2x \sin x + x^2 \cos x

$f' (x) = 2 x sin x + x 2 cos x$

c) Quotient Rule

f(x) = \frac{u(x)}{v(x)}

$f (x) = v (x) u (x)$ , then

f'(x) = \frac{u'v - uv'}{v^2}

$f' (x) = v 2 u' v - u v'$

Example:

f(x) = \frac{x^2}{x+1}

$f (x) = x + 1 x 2$ , then

f'(x) = \frac{(2x)(x+1) - x^2(1)}{(x+1)^2}

$f' (x) = (x + 1) 2 (2 x) (x + 1) - x 2 (1)$

d) Chain Rule

f(x) = g(h(x))

$f (x) = g (h (x))$ , then

f'(x) = g'(h(x)) \cdot h'(x)

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

Example:

f(x) = \sin(x^2)

$f (x) = sin (x 2)$ , then

f'(x) = \cos(x^2) \cdot 2x

$f' (x) = cos (x 2) \cdot 2 x$

3. Applications of Derivatives

a) Tangents and Normals

The tangent line to a function at $x = a$ $x = a$ has a slope given by $f'(a)$ f′(a).
The normal line is perpendicular to the tangent and has slope $-\frac{1}{f'(a)}$ −f′(a)1.
Used in engineering and physics to find slopes of curves.

b) Optimization (Maxima and Minima)

Derivatives help determine where a function reaches its highest (maximum) or lowest (minimum) points.
Critical Points: Solve $f'(x) = 0$ $f' (x) = 0$ to find these points.
Second Derivative Test:
- If $f''(x) > 0$ $f'' (x) > 0$ , the function has a local minimum.
- If $f''(x) < 0$ $f'' (x) < 0$ , the function has a local maximum.

Example: A company’s revenue function is

R(x) = -2x^2 + 40x

$R (x) = - 2 x 2 + 40 x$ . To maximize revenue, solve

R'(x) = 0

$R' (x) = 0$ :

R'(x) = -4x + 40 = 0 \Rightarrow x = 10

$R' (x) = - 4 x + 40 = 0 \Rightarrow x = 10$