Differentiation — Class 12 Maharashtra HSC Mathematics revision notes
A 5-page visual revision sheet for Differentiation, written for Class 12 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Differentiation in one look
the chain rule run in every direction: through a composite, backward through an inverse, sideways through an implicit or parametric pair
Derivative of a Composite Function
- a composite is a function of a function: sin(log x), e^tan x, log(x^5+4)
- chain rule: y=f(u), u=g(x), both differentiable, then dy/dx = (dy/du)(du/dx)
- three deep, y=f(v), v=g(u), u=h(x): dy/dx = (dy/dv)(dv/du)(du/dx)
- in practice: treat the inner block as one unit, differentiate the outer, multiply by its own derivative
- a product of composites, e.g. (x^3+2x-3)^4(x+cos x)^3: product rule first, chain rule inside each factor
- one factor at a time, u'v+uv', is where a chain-inside-a-product problem starts
Standard composite derivatives
- power of a function
d/dx[f(x)]^n = n[f(x)]^n-1f'(x) - sine of a function
d/dxsin[f(x)] = cos[f(x)]· f'(x) - exponential of a function
d/dxe^f(x) = e^f(x)· f'(x) - log of a function
d/dxlog[f(x)] = f'(x)/f(x)
Special cases in differentiation
- f(x) is a constant k — the derivative is zero, chain rule or not
- |x|=1 exactly, for sin^-1x or cos^-1x — the tangent is vertical, the derivative is undefined
- -1<x<1, for sec^-1x or cosec^-1x — the function itself is undefined there, not just its derivative
Chain rule through a square root
Q. Differentiate y=√x^2+5 w.r.t. x.
- treat x^2+5 as one block u, so y=√u
- d/dx[√u] = frac12√ucdotdu/dx, and du/dx = 2x
- dy/dx = frac12√x^2+5· 2x
Answer: dy/dx = x/√x^2+5
Chain rule stacked two deep
Q. Differentiate y=e^tan x w.r.t. x.
- outer function e^(·), inner block tan x
- d/dx[e^tan x] = e^tan xcdotd/dx(tan x)
- d/dx(tan x) = sec^2 x
Answer: dy/dx = e^tan xsec^2 x
Page 1 of 5. The full sheet, all 5 pages, opens free in the app.
What this Differentiation sheet covers
- Derivative of a Composite Function
- Standard composite derivatives
- power of a function
- sine of a function
- exponential of a function
- log of a function
- Special cases in differentiation
- Chain rule through a square root
- Chain rule stacked two deep
- Geometrical Meaning of the Derivative
- Derivative of an Inverse Function
- if y=f(x) is differentiable with dy/dx not 0, and x=f^-1(y) exists
- Derivative of an inverse function at a point
- Derivatives of Inverse Trigonometric Functions
- y=sin^-1x, |x|<1
- y=cos^-1x, |x|<1
- y=tan^-1x, any real x
- y=cot^-1x, any real x
- y=sec^-1x, x>1
- y=cosec^-1x, x>1
- Inverse trig via a trig substitution
- Substitutions worth memorising
- Logarithmic Differentiation
- for y=[f(x)]^g(x), take log of both sides first
- then differentiate implicitly and multiply back by y
- Logarithmic differentiation with a variable exponent
- Implicit Functions
- Implicit differentiation
- Derivatives of Parametric Functions
- x=f(t), y=g(t), both differentiable in the parameter t
- Parametric differentiation
- Implicit vs parametric differentiation
- Implicit
- Parametric
- One Function w.r.t. Another, and Higher Derivatives
- Second derivative of a polynomial
- nth derivative of standard functions
- power function, m a positive integer, m>=n
- reciprocal, linear
- log, linear
- sine, shifted
- cosine, shifted
- exponential, linear
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Differentiation the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.