Engneering Mathematics

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)=x2, then the derivative is

$f'(x) = 2x$

f′(x)=2x. 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

If

$f(x) = x^n$

f(x)=xn, then

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

f′(x)=nxn−1

Example:

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

f(x)=x3⇒f′(x)=3x2

b) Product Rule

If

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

f(x)=u(x)⋅v(x), then

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

f′(x)=u′v+uv′

Example:

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

f(x)=x2sinx, then

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

f′(x)=2xsinx+x2cosx

c) Quotient Rule

If

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

f(x)=v(x)u(x)​, then

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

f′(x)=v2u′v−uv′​

Example:

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

f(x)=x+1x2​, then

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

f′(x)=(x+1)2(2x)(x+1)−x2(1)​

d) Chain Rule

If

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

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

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

f′(x)=g′(h(x))⋅h′(x)

Example:

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

f(x)=sin(x2), then

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

f′(x)=cos(x2)⋅2x

---

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)=−2x2+40x. To maximize revenue, solve

$R'(x) = 0$

R′(x)=0:

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

R′(x)=−4x+40=0⇒x=10