Maharashtra State Board · Class 12 · Mathematics · Part II Chapter 1

Differentiation — Formula Sheet

Board Formulas

11 formulas

1.Definition (first principle)
$\frac{d y}{d x} = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$
Ab initio definition; used when derivative rules do not apply directly.
2.Sum, Product, Quotient Rules★
$(u + v)^{'} = u^{'} + v^{'}; (uv)^{'} = u^{'} v + u v^{'}; (\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}$
Fundamental algebra of derivatives. Do not confuse product and quotient rules.
3.Chain Rule★
$\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$
Differentiate outer function, keep inner unchanged, multiply by inner derivative.
4.Derivatives of Standard Functions★
$(x^{n})^{'} = n x^{n - 1}; (sin x)^{'} = cos x; (cos x)^{'} = - sin x; (e^{x})^{'} = e^{x}; (lo g x)^{'} = 1/ x$
Baseline table; memorise all six trigonometric derivatives and their inverses.
5.Inverse Trigonometric Derivatives
$\frac{d}{d x} (sin^{- 1} x) = \frac{1}{1 - x ^{2}}, \frac{d}{d x} (tan^{- 1} x) = \frac{1}{1 + x ^{2}}$
Note the domains: |x| < 1 for sin⁻¹; all real x for tan⁻¹.
6.Logarithmic Differentiation★
$y = f (x)^{g (x)} \Rightarrow ln y = g (x) ln f (x) \Rightarrow differentiate$
Take log both sides, then differentiate implicitly. Essential for x^x-type problems.
7.Implicit Differentiation★
$F (x, y) = 0 \Rightarrow \frac{d y}{d x} = - \frac{\partial F / \partial x}{\partial F / \partial y}$
Alternative: differentiate both sides w.r.t. x treating y as function of x; then solve for dy/dx.
8.Parametric Differentiation★
$x = f (t), y = g (t) \Rightarrow \frac{d y}{d x} = \frac{d y / d t}{d x / d t}$
Cancel dt; use chain rule. For d²y/dx² differentiate w.r.t. x again and use dt/dx.
9.Second-order Parametric Derivative
$\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d t} (\frac{d y}{d x}) \cdot \frac{d t}{d x}$
Do NOT just do (d²y/dt²)/(d²x/dt²) — that gives a wrong answer.
10.Higher-Order Derivatives
$\frac{d ^{n} y}{d x ^{n}} = (\frac{d}{d x})^{n} y$
Apply the derivative n times. Used in Taylor series, mean-value theorem.
11.Logarithm of a Product
$lo g (uv) = lo g u + lo g v, lo g (u / v) = lo g u - lo g v$
Useful in log-differentiation to simplify products/quotients before differentiating.