54  Differentiation

54.1 Introduction

Differentiation is a fundamental concept in calculus that deals with the rate of change of a function.
- If \(y=f(x)\), then the derivative \(\frac{dy}{dx}\) measures how \(y\) changes with a small change in \(x\).
- It is the foundation for slopes of curves, maxima–minima, optimization, velocity–acceleration, and growth–decay problems.

54.2 Geometric Meaning

  • Derivative = slope of tangent to curve \(y=f(x)\) at point \((x,y)\).
  • \(\frac{dy}{dx}>0\) → function is increasing.
  • \(\frac{dy}{dx}<0\) → function is decreasing.
  • \(\frac{dy}{dx}=0\) → horizontal tangent (possible extremum).

54.3 Physical Meaning

  • If \(s=f(t)\) is displacement, then \(\frac{ds}{dt}=\) velocity, \(\frac{d^2s}{dt^2}=\) acceleration.
  • In economics: marginal cost, marginal revenue, elasticity use derivatives.

54.4 Rules of Differentiation

54.4.1 1. Basic Derivatives

  1. \(\frac{d}{dx}(c)=0\) (constant)
  2. \(\frac{d}{dx}(x^n)=nx^{n-1}\)
  3. \(\frac{d}{dx}(e^x)=e^x\)
  4. \(\frac{d}{dx}(a^x)=a^x \ln a\)
  5. \(\frac{d}{dx}(\ln x)=\tfrac{1}{x}\)
  6. \(\frac{d}{dx}(\sin x)=\cos x\)
  7. \(\frac{d}{dx}(\cos x)=-\sin x\)
  8. \(\frac{d}{dx}(\tan x)=\sec^2x\)
  9. \(\frac{d}{dx}(\cot x)=-\csc^2x\)
  10. \(\frac{d}{dx}(\sec x)=\sec x \tan x\)
  11. \(\frac{d}{dx}(\csc x)=-\csc x \cot x\)

54.4.2 2. Rules

  • Sum Rule: \((f+g)'=f'+g'\)
  • Product Rule: \((uv)'=u'v+uv'\)
  • Quotient Rule: \(\Big(\frac{u}{v}\Big)'=\frac{u'v-uv'}{v^2}\)
  • Chain Rule: \(\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}\)

54.4.3 3. Higher Order Derivatives

  • Second derivative: \(\frac{d^2y}{dx^2}\) → concavity/convexity.
  • Third, fourth derivatives used in physics and advanced modeling.

54.5 Maxima and Minima

To find turning points of \(y=f(x)\):
1. Find \(f'(x)=0\).
2. Check \(f''(x)\):
- \(f''(x)>0\) → local minimum.
- \(f''(x)<0\) → local maximum.
- \(f''(x)=0\) → test further (point of inflection possible).

54.6 Differentiation of Special Functions

54.6.1 Exponential and Logarithmic

  • \(\frac{d}{dx}(e^{kx})=ke^{kx}\)
  • \(\frac{d}{dx}(\ln(ax+b))=\frac{1}{ax+b}\cdot a\)

54.6.2 Trigonometric

  • \(\frac{d}{dx}(\sin^2x)=2\sin x \cos x=\sin 2x\)
  • \(\frac{d}{dx}(\tan^{-1}x)=\tfrac{1}{1+x^2}\)
  • \(\frac{d}{dx}(\sin^{-1}x)=\tfrac{1}{\sqrt{1-x^2}}\)
  • \(\frac{d}{dx}(\cos^{-1}x)=-\tfrac{1}{\sqrt{1-x^2}}\)

54.7 Applications

54.7.1 1. Slopes and Tangents

Equation of tangent at \((x_0,y_0)\):
\(y-y_0=f'(x_0)(x-x_0)\).

Equation of normal: slope = \(-\tfrac{1}{f'(x_0)}\).

54.7.2 2. Increasing/Decreasing Functions

  • If \(f'(x)>0\) on interval → increasing.
  • If \(f'(x)<0\) → decreasing.

54.7.3 3. Maxima–Minima Optimization

  • Used in business (maximize profit, minimize cost).
  • In physics (maximize height, minimize distance).

54.7.4 4. Approximation

For small \(h\):
\(f(x+h)\approx f(x)+hf'(x)\).

54.8 Solved Examples

Example 1
Differentiate \(y=5x^3-2x^2+7\).
\(y'=15x^2-4x\).

Example 2
Find \(\frac{dy}{dx}\) if \(y=\ln(x^2+1)\).
\(y'=\frac{2x}{x^2+1}\).

Example 3
Find slope of tangent to \(y=x^2\) at \((2,4)\).
\(y'=2x \Rightarrow\) slope = 4.
Equation: \(y-4=4(x-2)\Rightarrow y=4x-4\).

Example 4
Find local maxima/minima of \(f(x)=x^2-4x+3\).
\(f'(x)=2x-4=0 \Rightarrow x=2\).
\(f''(x)=2>0 \Rightarrow\) local minimum at \((2,-1)\).

Example 5
Differentiate \(y=\tan^{-1}\frac{2x}{1-x^2}\).
Use substitution: \(y=\tan^{-1}(\tan 2x)=2x\).
\(\Rightarrow y'=2\).

Example 6
Equation of tangent to \(y=\sqrt{x}\) at \((4,2)\).
\(y'=\frac{1}{2\sqrt{x}}\). At \(x=4\), slope=\(1/4\).
Equation: \(y-2=\tfrac{1}{4}(x-4)\).

54.9 Practice Problems

  1. Differentiate \(y=3x^4+5x-7\).
  2. Find \(\frac{dy}{dx}\) if \(y=\ln(\sin x)\).
  3. Find slope of tangent to \(y=x^3\) at \((1,1)\).
  4. Differentiate \(y=e^{2x}\sin x\).
  5. If \(y=\frac{x^2+1}{x}\), find \(\frac{dy}{dx}\).
  6. Find maxima/minima of \(y=x^3-6x^2+9x+15\).
  7. Find equation of tangent to \(y=\cos x\) at \(x=\pi/3\).
  8. Differentiate \(y=\sin^{-1}(2x\sqrt{1-x^2})\).
  9. Find \(f''(x)\) if \(f(x)=\ln(1+x)\).
  10. Approximate \(\sqrt{101}\) using derivatives.

54.10 Answer Key (Concise)

  1. \(12x^3+5\)
  2. \(\cot x\)
  3. Slope = 3, tangent: \(y-1=3(x-1)\)
  4. \(e^{2x}(2\sin x+\cos x)\)
  5. \((x^2-1)/x^2\)
  6. \(f'(x)=3x^2-12x+9=0 \Rightarrow x=1,3\); maxima at \((1,19)\), minima at \((3,15)\)
  7. \(y'=-\sin x\), slope=\(-\sqrt{3}/2\), tangent: \(y-\tfrac{1}{2}=(-\sqrt{3}/2)(x-\pi/3)\)
  8. Differentiate using chain rule → \(\frac{2\sqrt{1-x^2}-2x\cdot\frac{x}{\sqrt{1-x^2}}}{\sqrt{1-(2x\sqrt{1-x^2})^2}}\) (simplify)
  9. \(f'(x)=1/(1+x)\), \(f''(x)=-1/(1+x)^2\)
  10. \(\sqrt{100+h}\approx10+\frac{h}{20}\), \(h=1\) → \(\approx10.05\)

54.11 Summary

  • Differentiation = rate of change, slope of tangent.
  • Rules: power rule, product, quotient, chain rule.
  • Applications: slope, tangents, maxima–minima, optimization.
  • Higher derivatives indicate concavity and acceleration.
  • Essential for calculus-based aptitude and real-world applications.