Differentiation from the First Principles

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Analytical Geometry
Circles Equation given center and radius given ends of diameter General equation finding center and radius Standard forms of equations of Parabola Standard forms of equations of Ellipse Standard forms of equations of Hyperbola simple properties
Complex numbers
modulus and conjugate Arithmetic operations on complex number Modulus Amplitude form (Polar form) Euler form (exponential form) Properties De Movire’s Theorem and its applications
Differential Equations
Definition of a differential equation Order and Degree Formation of differential equations First order of the differential equations First degree of the differential equations Variable separable Homogeneous equations Liner differential equations form Bernoulli's equations nth order linear differential equation constant Finding Particular Integrals for functions
Diffrentiation and it's Applications
Functions and limits Standard Limits Diffrentiation of Sum Diffrentiation of Product Quotient Of Functions Trigonometric Inverse Trigonometric Differentiation from the First Principles Exponential logarithmic Function of a Function Hyperbolic Functions implicit Explicit and Parametric Functions Derevative Function with respect to another function Second order derivatives Geometrical Applications of the Derivative Increasing and decreasing functions Derivative as a Rate Measure Partial Differentiation maxima and minima using second order derevative only Partial derivatives up to second order Euler’s theorem Errors and Approximations
Integration and Its Applications
Indefinite Integral Standard forms Integration by decomposition of the integrand of trigonometric algebraic exponential logarithmic and Hyperbolic functions Integration by substitution Integration of reducible and irreducible quadratic factors Integration by part Definite Integrals and properties Definite Integral as the limit of a sum Application of Integration to find areas under plane curves and volumes of Solids of revolution Mean and RMS value
Matrices
Matrices of 3rd order Types of matrices Algebra of matrices Transpose of matrices Symmetric Skew symmetric matrices Minor Cofactor of an element Determinant of a square matrix Properties Laplace's expansion Singular and non singular matrices Adjoint matrices Multiplicative inverse of a square matrix System of linear equations in 3 variables Cramer's rule Matrix inversion method
Partial fractions
Resolving a given rational function into partial fractions.
TRIGONOMETRY
Properties of Trigonometric functions Ratios of compound angles Multiple angles Sub multiple angles Transformations of Products into sum or difference and vice versa Simple trigonometric equations Properties of triangles Inverse Trigonometric functions
Find the derivative of f(x)=x² using first principles.
A
2x
B
x
C
x²
D
1
Analysis & Theory

Using first principle: f'(x)=lim (h→0) [(x+h)² - x²]/h = lim (h→0) [2xh + h²]/h = 2x + h. As h→0, derivative = 2x.