Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Algebra of Derivatives

Let  u, v are two derivable functions of  x .

(i)  (u + v)' = u'+ v'

(ii)  (u - v)' = u'- v'

(iii)  (u \cdot v)' = u'v + uv'

(iv)  \left( \dfrac{u}{v} \right)' = \dfrac{v u' - uv'}{v^2},~~ [v(x) \neq 0]

Tangent Line of a Graph at a Point

Slope of tangent i.e.  \tan \theta = \dfrac{\delta y}{\delta x} = \dfrac{f(x + \delta x) - f(x)}{\delta x}

 

 f  is continous at  x , i.e.  f(x + \delta x) \to f(x) as  \delta x \to 0 i.e.  (x + \delta x, y + \delta y) \longrightarrow (x, y) as  \deltax \to 0 i.e  \lim_{\delta x \to 0} \dfrac{\delta y}{\delta x} exists,  \implies \dfrac{dy}{dx} = f'(x) where  \overleftrightarrow {PQ} is called the tangent line to the curve at  P .

Definition:

The tangent line of a function  f at the point  P = (x_0, f(x_0)) is:

(i) The line through  P with slope  f'(x_0) if  f^1(x)_0) =\tan \phi .

(ii) The line  x = x_0 if  \lim_{x \to n_0} \left| \dfrac{f(x) - f(x_0)}{x-x_0} \right| = \infty .

It holds when  y - f(x_0) = f'(x_0)(x - x_0)

Indeterminant forms:

 \dfrac{0}{0}, \dfrac{\infty}{\infty}, 0 \times \infty, 1^{\infty}, 0^0, \infty^0, \infty - \infty

L’ Hospital’s Rule

It is pronounced “lopital”. The limit when divide one function by another is the same after we take the derivative of each function, i.e.  \lim_{x \to a} \dfrac{f(x)}{g(x)} = \lim_{x \to a} \dfrac{f'(x)}{g'(x)}

Note:

(i)  \infty + \infty = \infty

(ii)  \infty \times \infty = \infty

(iii)  \dfrac{a}{\infty} = 0 , if  a is finite.

(iv)  \dfrac{a}{0}   is not defined, if   a \neq 0 .

(v)  ab = 0, iff  a = 0 or   b = 0 and   a, b are finite.

 

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