Lecture-7

Some Important Results of Derivatives:

\frac{d}{dx} (x) = 1
\frac{d}{dx} (constant) = 0
\frac{d}{dx} (x^n) = n{x}^{n-1}
\frac{d}{dx} \left ( \frac{1}{x^n} \right ) = \frac{-n}{x^{n+1}}

\frac{d}{dx} (\log x) = \frac{1}{x}
\frac{d}{dx} (e^x) = e^x
\frac{d}{dx} (a^x) = {a^x} \log a
\frac{d}{dx} (\sqrt{x}) = \frac{1}{2\sqrt{x}}

\frac{d}{dx} (\sin x) = \cos x
\frac{d}{dx} (\cos x) = – \sin x
\frac{d}{dx} (\tan x) = \sec^{2} x
\frac{d}{dx} (\sec x) = \sec x . \tan x
\frac{d}{dx} (\cosec x) = – \cosec x . \cot x
\frac{d}{dx} (\cot x) = – \cosec^2 x

\frac{d}{dx} (sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}
\frac{d}{dx} (cos^{-1}x) = -\frac{1}{\sqrt{1-x^2}}

\frac{d}{dx} (tan^{-1}x) = \frac{1}{1+x^2}
\frac{d}{dx} (cot^{-1}x) = -\frac{1}{1+x^2}
\frac{d}{dx} (sec^{-1}x) = \frac{1}{x{\sqrt{x^2-1}}}
\frac{d}{dx} (cosec^{-1}x) = -\frac{1}{x{\sqrt{x^2-1}}}

PRODUCT RULE:
(I \times II)’ = I (II)’ + II (I)’

QUOTIENT RULE:
\left( \frac{N}{D} \right )^’ = \frac{D(N)’ – N(D)’}{D^2}

LOGARITHM RULES:
\log (m \times n) = \log m + \log n
  \log \left ( \frac{m}{n} \right ) = \log m – \log n
\log (m)^n = n \log m
\log_a (b) = \frac{\log_x (b)}{\log_x (a)}
\log_e(e^x) = x
\log_a(a^x) = x
(e)^{\log_e{x}} = x
(a)^{\log_a{x}} = x

Continuity and Differentiability Lecture 7
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In this lecture, I am discussing remaining questions from NCERT Exercise 5.5 which are based on logarithmic differentiation.

Questions discussed in this lecture:

NCERT EXERCISE 5.5 (Logarithmic Differentiation)

Differentiate the functions given in Exercises 1 to 11 w.r.t. x.

Question 4. x^x – 2^{\sin x}

Question 5. (x+3)^2.(x+4)^3.(x+5)^4

Question 6. \left( x + \frac{1}{x} \right)^x + x^{\left( 1 + \frac{1}{x} \right) }

Question 7. (\log x)^x + x^{\log x}

Question 8. (\sin x)^x + \sin^{-1}{\sqrt{x}}

Question 9. x^{\sin x} + (\sin x)^{\cos x}

Question 10. x^{x \cos x} + \frac{x^2 + 1}{x^2 – 1}

Question 11. (x \cos x)^x + (x \sin x)^{\frac{1}{x}}

मत सोच की तेरा सपना क्यों पूरा नहीं होता,
हिम्मत वालों का इरादा कभी अधूरा नहीं होता,
जिस इंसान के कर्म अच्छे होते है,
उसके जीवन में कभी अँधेरा नहीं होता।