Applications of Derivatives

1. Application of Derivative in Geometry

• Equations of Tangent and Normal

  Let $y=f(x)$ be a continuous function representing a curve, and $P(x_1, y_1)$ be a point on it.

  Slope of Tangent ($m$):

  $$ m = \left[\frac{dy}{dx}\right]_{(x_1, y_1)} $$

  Equation of Tangent: $y - y_1 = m(x - x_1)$

  Slope of Normal ($m'$): The normal is perpendicular to the tangent, so:

  $$ m' = -\frac{1}{\left[\frac{dy}{dx}\right]_{(x_1, y_1)}} \quad \left(\text{if } \left[\frac{dy}{dx}\right]_{(x_1, y_1)} \neq 0\right) $$

  Equation of Normal: $y - y_1 = m'(x - x_1)$

2. Derivative as a Rate Measure

• Rates of Change

  Average rate of change: $\frac{\delta y}{\delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$ (Geometrically, the slope of the secant line).

  Instantaneous rate of change: $\lim_{\delta x \to 0}\left(\frac{\delta y}{\delta x}\right) = \frac{dy}{dx}$

• Kinematics (Motion in a straight line)

  Let $s = f(t)$ be the displacement of a particle at time $t$.

  Velocity ($v$): Rate of change of displacement.

  $$ v = f'(t) = \frac{ds}{dt} $$

  Speed: Absolute value of velocity, $|v| = |f'(t)|$

  Acceleration ($a$): Rate of change of velocity.

  $$ a = \frac{dv}{dt} = \frac{d^2s}{dt^2} $$

  Jerk ($j$): Rate of change of acceleration.

  $$ j = \frac{da}{dt} = \frac{d^3s}{dt^3} $$

3. Approximations

• Linear Approximation Formula

  For a sufficiently small increment $h$, the approximate value of the function at $x = a+h$ is given by:

  $$ f(a+h) \approx f(a) + hf'(a) $$

4. Rolle's Theorem (Rolle's Lemma)

• Statement of Rolle's Theorem

  If a real-valued function $f$ satisfies three conditions:

  1. It is continuous on the closed interval $[a, b]$.
  2. It is differentiable on the open interval $(a, b)$.
  3. $f(a) = f(b)$.

  Then there exists at least one point $c \in (a, b)$ such that $f'(c) = 0$.

  Geometrical Meaning: The tangent to the curve at $x=c$ is parallel to the X-axis.

  Geometrical Significance of Rolle's Theorem: $f'(c) = 0$

5. Lagrange's Mean Value Theorem (LMVT)

• Statement of LMVT

  If a real-valued function $f$ satisfies two conditions:

  1. It is continuous on the closed interval $[a, b]$.
  2. It is differentiable on the open interval $(a, b)$.

  Then there exists at least one point $c \in (a, b)$ such that:

  $$ f'(c) = \frac{f(b) - f(a)}{b - a} $$

  Geometrical Meaning: The tangent to the curve at $x=c$ is parallel to the secant chord AB joining the endpoints.

  Geometrical Significance of LMVT: Tangent parallel to Chord

6. Increasing and Decreasing Functions

• Monotonicity Rules

  Strictly Increasing: A function $f$ is strictly increasing on $(a,b)$ if $f'(x) > 0$.

  Strictly Decreasing: A function $f$ is strictly decreasing on $(a,b)$ if $f'(x) < 0$.

  Monotonically Increasing: $f'(x) \ge 0$.

  Monotonically Decreasing: $f'(x) \le 0$.

7. Maxima and Minima

• First Derivative Test

  A function $f(x)$ has a local maxima at $x=c$ if:

  1. $f'(c) = 0$ (Stationary point)
  2. $f'(c-h) > 0$ (Increasing before $c$)
  3. $f'(c+h) < 0$ (Decreasing after $c$)

  A function $f(x)$ has a local minima at $x=c$ if:

  1. $f'(c) = 0$ (Stationary point)
  2. $f'(c-h) < 0$ (Decreasing before $c$)
  3. $f'(c+h) > 0$ (Increasing after $c$)

  Point of Inflexion: If $f'(c)=0$ but the sign of the derivative does not change across $c$.

• Second Derivative Test

  Let $f(x)$ be twice differentiable at $x=c$ and $f'(c) = 0$:

  • If $f''(c) < 0$, then $f(x)$ has a maxima at $x=c$.
  • If $f''(c) > 0$, then $f(x)$ has a minima at $x=c$.
  • If $f''(c) = 0$, the test fails (use the First Derivative Test).
  Tangents behavior around Local Maxima and Minima