निगाहों में मंजिल थी,
गिरे और गिरकर संभलते रहे,
हवाओं ने बहुत कोशिश की,
मगर चिराग आँधियों में भी जलते रहे।

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)

Find \( \frac{dy}{dx} \) of the functions given in Exercises 12 to 15.

Question 12. \( x^y + y^x =1 \)

Question 13. \( y^x = x^y \)

Question 14. \( (\cos x)^y = (\cos y)^x \)

Question 15. \( xy = e^{(x-y)} \)

Question 16. Find the derivative of the function given by \( f(x) = (1+x)(1+x^2)(1+x^4)(1+x^8) \) and hence find f′(1).

Question 17. Differentiate \( (x^2-5x+8)(x^3+7x+9) \) in three ways mentioned below:
(i) by using product rule
(ii) by expanding the product to obtain a single polynomial.
(iii) by logarithmic differentiation.
Do they all give the same answer?

Question 18. If u, v and w are functions of x, then show that \( \frac{d}{dx}(u.v.w) = \frac{du}{dx}.v.w + u.\frac{dv}{dx}.w + u.v.\frac{dw}{dx} \) in two ways – first by repeated application of product rule, second by logarithmic differentiation.