Class: 12th
Subject: Mathematics
Chapter: Differentiation of Sec-E
Writer and Publisher: S. Chand & OP Malhotra Publications
Exercise 8e
Formula:
- Differentiation Formula
d/dx (xⁿ) = n·xⁿ⁻¹
d/dx (ax+b)ⁿ = n·(ax+b)ⁿ⁻¹ · d/dx (ax+b)
d/dx (a constant) = 0
d/dx [f(x)+c] = d/dx [f(x)]
d/dx [k(f(x))] = k · d/dx [f(x)]
d/dx (u+v) = du/dx + dv/dx
d/dx (u+v+w+…) = du/dx + dv/dx + dw/dx + …
d/dx (uv) = u·dv/dx + v·du/dx
d/dx (u/v) = (v·du/dx – u·dv/dx) / v²
d/dx (x) = 1
d/dx (1/x) = -1/x²
d/dx (√x) = 1/(2√x)
d/dx (1/x²) = -2/x³
d/dx (1/x³) = -3/x⁴
- Logarithm, Half Angle & Double Angle Formulas
logarithm functions:
log mⁿ = n·log m
log mn = log m + log n
log m/n = log m – log n
log eˣ = x
log a = 0
log e = 1
xˣ = e^(x logx)
1/2 logx = log root x
Half angle formula:
Sin(θ/2) = ±√((1-cosθ)/2)
Cos(θ/2) = ±√((1+cosθ)/2)
tan(θ/2) = (1-cosθ)/sinθ = sinθ/(1+cosθ) = ±√((1-cosθ)/(1+cosθ))
Double angle formula
Sin 2θ = 2·sinθ·cosθ = 2tanθ/(1+tan²θ)
Cos 2θ = cos²θ – sin²θ = 1-2sin²θ = 2cos²θ – 1
tan 2θ = 2tanθ/(1-tan²θ)
cos²θ = (1-tan²θ)/(1+tan²θ)
(Side notes)
sin(any angle) = ±√((1-cos(double angle))/2)
tan(any angle) = √((1-cos(double angle))/(1+cos(double angle)))
- More Differentiation Formulas
d/dx (sinx) = cosx
d/dx (cosx) = -sinx
d/dx (tanx) = sec²x
d/dx (cosec x) = -cosec x · cot x
d/dx (sec x) = sec x · tan x
d/dx (cot x) = -cosec²x
d/dx (aˣ) = aˣ · log a
d/dx (eˣ) = eˣ
d/dx (log x) = 1/x
d/dx (logₐ x) = 1/(x·log a)
Differentiation of Inverse Trigonometric function
d/dx (sin⁻¹x) = 1/√(1-x²)
d/dx (cos⁻¹x) = -1/√(1-x²)
d/dx (tan⁻¹x) = 1/(1+x²)
d/dx (cosec⁻¹x) = -1/(|x|√(x²-1))
d/dx (sec⁻¹x) = 1/(|x|√(x²-1))
d/dx (cot⁻¹x) = -1/(1+x²)
- Triple Angle, Sum & Difference Formulas
Triple angle formula
sin 3θ = 3sinθ – 4sin³θ
cos 3θ = 4cos³θ – 3cosθ
tan 3θ = (3tanθ – tan³θ) / (1-3tan²θ)
Sum formula:
sin(A+B) = sinAcosB + cosAsinB
cos(A+B) = cosAcosB – sinAsinB
tan(A+B) = (tanA + tanB) / (1 – tanA·tanB)
Difference formula:
sin(A-B) = sinAcosB – cosAsinB
cos(A-B) = cosAcosB + sinAsinB
tan(A-B) = (tanA – tanB) / (1 + tanA·tanB)
