JEE Mathematics Application of Derivatives

Application Of Derivatives Formulas

Equation of tangent and normal:
Tangent at \( (x_1, y_1) \) is given by \( (y - y_1) = f'(x_1)(x - x_1) \); when, \( f'(x_1) \) is real.
And normal at \( (x_1, y_1) \) is \( (y - y_1) = -\frac{1}{f'(x_1)}(x - x_1) \), when \( f'(x_1) \) is nonzero real.

Tangent from an external point:
Given a point \( P(a, b) \) which does not lie on the curve \( y = f(x) \), then the equation of possible tangents to the curve \( y = f(x) \), passing through \( (a, b) \) can be found by solving for the point of contact Q.
\( f'(h) = \frac{f(h) - b}{h - a} \) and equation of tangent is \( y - b = \frac{f(h)-b}{h-a}(x-a) \).

Length of tangent, normal, subtangent, subnormal:
(i) \( \text{PT} = |k| \sqrt{1 + \frac{1}{m^2}} = \text{Length of Tangent} \)
(ii) \( \text{PN} = |k| \sqrt{1 + m^2} = \text{Length of Normal} \)
(iii) \( \text{TM} = \left| \frac{k}{m} \right| = \text{Length of subtangent} \)
(iv) \( \text{MN} = |km| = \text{Length of subnormal} \).

Angle between the curves:
\( \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right| \)

Rolle’s Theorem :
If a function \( f \) defined on \( [a, b] \) is (i) continuous on \( [a, b] \), (ii) derivable on \( (a, b) \) and (iii) \( f(a) = f(b) \), then there exists at least one real number \( c \) between \( a \) and \( b (a < c < b) \) such that \( f'(c) = 0 \).

Lagrange’s Mean Value Theorem (LMVT) :
If a function \( f \) defined on \( [a, b] \) is (i) continuous on \( [a, b] \) and (ii) derivable on \( (a, b) \), then there exists at least one real numbers \( c \) between \( a \) and \( b (a < c < b) \) such that \( \frac{f(b) - f(a)}{b - a} = f'(c) \).