Differentiation — JEE Mains Mathematics

Formulas (2)

• Simplify before you differentiate · A standard substitution
  $$x=\tan\theta:\quad\tan^{-1}\frac{2x}{1-x^2}=2\theta=2\tan^{-1}x\quad(|x|<1)$$
• The chain rule and inverse functions · Derivative of an inverse
  $$g'(k)=\frac{1}{f'(a)},\qquad f(a)=k$$

Watch out for (2)

• The branch decides the answer
  → Simplify before you differentiate
• Invert at the right point
  → The chain rule and inverse functions

Formulas (2)

• Implicit and parametric differentiation · Second derivative, parametric
  $$\frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\Big/\frac{dx}{dt}$$
• Logarithmic differentiation · Variable power
  $$\frac{d}{dx}f^{\,g}=f^{\,g}\left(g'\ln f+\frac{g\,f'}{f}\right)$$

Watch out for (2)

• Not the ratio of second derivatives
  → Implicit and parametric differentiation
• d²x/dy² is not 1/y″
  → Logarithmic differentiation

Formulas (2)

• Functional equations · Derivative from the equation
  $$f(x+y)=f(x)f(y)\ \Rightarrow\ f'(x)=\lim_{h\to0}f(x)\frac{f(h)-1}{h}=f'(0)\,f(x)$$
• Derivative values as constants · For a cubic
  $$f(x)=x^3+ax^2+bx+c\ \Rightarrow\ f'(x)=3x^2+2ax+b,\quad f''(x)=6x+2a,\quad f'''(x)=6$$

Watch out for (2)

• Find f(0) first
  → Functional equations
• The coefficient is not a function
  → Derivative values as constants

Formulas (3)

• Where two pieces join · Smooth join at x = a
  $$f_1(a)=f_2(a)\quad\text{and}\quad f_1'(a)=f_2'(a)$$
• Composites and running maxima · Chain rule at a point
  $$(g\circ f)'(a)=g'\big(f(a)\big)\,f'(a)\quad\text{when both derivatives exist}$$
• The derivative from its definition · Derivative at a point
  $$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$

Watch out for (3)

• Continuity comes first
  → Where two pieces join
• A bad inner function can be smoothed out
  → Composites and running maxima
• f′ can exist without being continuous
  → The derivative from its definition

Formulas (2)

• Corners of modulus, max and min · Slope jump at a corner
  $$k\,|x-a|:\quad f'(a^+)-f'(a^-)=2k$$
• Jumps of the greatest integer function · Greatest integer function
  $$[x]=n\ \text{ for }\ n\le x<n+1,\qquad[x+n]=[x]+n\ \ (n\in\mathbb{Z})$$

Watch out for (2)

• A candidate is not yet a corner
  → Corners of modulus, max and min
• A jump counts twice in m + n
  → Jumps of the greatest integer function