If x y = y x then d y d x = y ( x log y - y ) x ( y log x - x ) .

Given, x y = y x Applying log on both sides, log x y = log y x ⇒ y log x = x log y Differentiating w.r.t x , we get d d x ( y log x ) = d d x ( x log y ) ⇒ y · 1 x + log x · d y d x = x 1 y · d y d x + log y ⇒ log x · d y d x - x y · d y d x = log y - y x ⇒ log x - x y d y d x = log y - y x ⇒ y log x - x y · d y d x = x log y - y x ∴ d y d x = y ( x log y - y ) x ( y log x - x ) Hence, proved.

If x y = y x then d y d x = y ( x log y - y ) x ( y log x - x ) .

Given, x y = y x Applying log on both sides, log x y = log y x ⇒ y log x = x log y Differentiating w.r.t x , we get d d x ( y log x ) = d d x ( x log y ) ⇒ y · 1 x + log x · d y d x = x 1 y · d y d x + log y ⇒ log x · d y d x - x y · d y d x = log y - y x ⇒ log x - x y d y d x = log y - y x ⇒ y log x - x y · d y d x = x log y - y x ∴ d y d x = y ( x log y - y ) x ( y log x - x ) Hence, proved.

If x 2 3 + y 2 3 = a 2 3 , then prove that d y d x = - y x 3 .

Given, x 2 3 + y 2 3 = a 2 3 Differentiating w.r.t x , we get d d x x 2 3 + y 2 3 = d d x a 2 3 ⇒ 2 3 x - 1 3 + 2 3 y - 1 3 · d y d x = 0 ⇒ 2 3 · 1 y 1 3 · d y d x = - 2 3 · 1 x 1 3 ⇒ d y d x = - y 1 3 x 1 3 ∴ d y d x = - y x 3 Hence, proved.

Find d y d x of the function y = ( 1 - 2 x ) 2 3 ( 1 + 3 x ) - 3 4 ( 1 - 6 x ) 5 6 ( 1 + 7 x ) - 6 7 . Find $\frac{d y}{d x}$ of each of the following functions.Find $\frac{d y}{d x}$ of each of the following functions.Find $\frac{d y}{d x}$ of each of the following functions.

Given, y = ( 1 - 2 x ) 2 3 ( 1 + 3 x ) - 3 4 ( 1 - 6 x ) 5 6 ( 1 + 7 x ) - 6 7 Applying log on both sides, log y = log ( 1 - 2 x ) 2 3 ( 1 + 3 x ) - 3 4 ( 1 - 6 x ) 5 6 ( 1 + 7 x ) - 6 7 = log ( 1 - 2 x ) 2 3 + log ( 1 + 3 x ) - 3 4 - log ( 1 - 6 x ) 5 6 - log ( 1 + 7 x ) - 6 7 = 2 3 log ( 1 - 2 x ) - 3 4 · log ( 1 + 3 x ) - 5 6 log ( 1 - 6 x ) + 6 7 log ( 1 + 7 x ) Differentiating w.r.t x , we get d d x log y = d d x 2 3 log ( 1 - 2 x ) - 3 4 · log ( 1 + 3 x ) - 5 6 log ( 1 - 6 x ) + 6 7 log ( 1 + 7 x ) ⇒ 1 y d y d x = 2 3 × 1 ( - 2 ) 1 - 2 x - 3 4 × 1 × 3 1 + 3 x - 5 6 × 1 1 - 6 x × - 6 + 6 7 × 1 1 + 7 x × 7 ⇒ 1 y · d y d x = - 4 3 ( 1 - 3 x ) - 9 4 ( 1 + 3 x ) + 5 1 - 6 x + 6 1 + 7 x ∴ d y d x = y 5 1 - 6 x + 6 1 + 7 x - 4 3 ( 1 - 2 x ) - 9 4 ( 1 + 2 x )

Find d y d x of the function y = x 4 · x 2 + 4 3 4 x 2 - 7 . Find $\frac{d y}{d x}$ of each of the following functions.Find $\frac{d y}{d x}$ of each of the following functions.Find $\frac{d y}{d x}$ of each of the following functions.

Given, d d x x 4 x 2 + 4 3 4 x 2 - 7 = d d x x 4 x 2 + 4 3 · 4 x 2 - 7 - x 4 x 2 + 4 3 · d d x 4 x 2 - 7 4 x 2 - 7 2 = d d x x 4 · x 2 + 4 3 + x 4 · d d x x 2 + 4 3 4 x 2 - 7 - x 4 x 2 + 4 3 · 1 2 4 x 2 - 7 1 2 - 1 · d d x 4 x 2 - 7 4 x 2 - 7 = 4 x 3 x 2 + 4 3 + x 4 · 1 3 x 2 + 4 1 3 - 1 · d d x x 2 + 4 4 x 2 - 7 - x 4 x 2 + 4 3 4 · d d x x 2 + d d x [ - 7 ] 2 4 x 2 - 7 4 x 2 - 7 = 4 x 3 x 2 + 4 3 + x 4 d d x x 2 + d d x [ 4 ] 3 x 2 + 4 2 3 4 x 2 - 7 - x 4 x 2 + 4 3 ( 4 · 2 x + 0 ) 2 4 x 2 - 7 4 x 2 - 7 = 4 x 3 x 2 + 4 3 + x 4 ( 2 x + 0 ) 3 x 2 + 4 2 3 4 x 2 - 7 - 4 x 5 x 2 + 4 3 4 x 2 - 7 4 x 2 - 7 = 4 x 2 - 7 4 x 3 x 2 + 4 3 + 2 x 5 3 x 2 + 4 2 3 - 4 x 5 x 2 + 4 3 4 x 2 - 7 4 x 2 - 7 ∴ d d x x 4 x 2 + 4 3 4 x 2 - 7 = 4 x 2 - 7 4 x 3 x 2 + 4 3 + 2 x 5 3 x 2 + 4 2 3 - 4 x 5 x 2 + 4 3 4 x 2 - 7 4 x 2 - 7

Find d y d x of the function y = ( a - x ) 2 ( b - x ) 3 ( c - 2 x ) 3 . Find $\frac{d y}{d x}$ of each of the following functions.Find $\frac{d y}{d x}$ of each of the following functions.Find $\frac{d y}{d x}$ of each of the following functions.

Given, d d x ( a - x ) 2 ( b - x ) 3 ( c - 2 x ) 3 = d d x ( a - x ) 2 ( b - x ) 3 · ( c - 2 x ) 3 - a - x 2 b - x 3 · d d x ( c - 2 x ) 3 ( c - 2 x ) 3 2 = d d x ( a - x ) 2 · b - x 3 + a - x 2 · d d x ( b - x ) 3 c - 2 x 3 - a - x 2 b - x 3 · 3 c - 2 x 2 d d x c - 2 x ( c - 2 x ) 6 = 2 a - x · d d x [ a - x ] · b - x 3 + a - x 2 · 3 b - x 2 · d d x [ b - x ] c - 2 x 3 - a - x 2 b - x 3 · 3 c - 2 x 2 d d x c - 2 · d d x [ x ] ( c - 2 x ) 6 = 6 c - 2 x 2 a - x 2 b - x 3 + 2 ( a - x ) ( 0 - 1 ) ( b - x ) 3 + ( a - x ) 2 · 3 ( b - x ) 2 ( 0 - 1 ) c - 2 x 3 ( c - 2 x ) 6 ∴ d d x ( a - x ) 2 ( b - x ) 3 ( c - 2 x ) 3 = 6 c - 2 x 2 a - x 2 b - x 3 + - 2 ( a - x ) ( b - x ) 3 - 3 ( a - x ) 2 ( b - x ) 2 ( c - 2 x ) 3 ( c - 2 x ) 6

Find d y d x of the function y = x 3 · 2 + 3 x ( 2 + x ) ( 1 - x ) . Find $\frac{d y}{d x}$ of each of the following functions.Find $\frac{d y}{d x}$ of each of the following functions.Find $\frac{d y}{d x}$ of each of the following functions.

Given, d d x x 3 3 x + 2 ( 1 - x ) ( x + 2 ) = d d x x 3 3 x + 2 · 1 - x x + 2 - x 3 3 x + 2 · d d x 1 - x x + 2 ( ( 1 - x ) ( x + 2 ) ) 2 = d d x x 3 · 3 x + 2 + x 3 · d d x [ 3 x + 2 ] 1 - x x + 2 - x 3 3 x + 2 d d x 1 - x · x + 2 + 1 - x · d d x x + 2 ( 1 - x ) 2 ( x + 2 ) 2 = 3 x 2 3 x + 2 + x 3 · 1 2 3 x + 2 1 2 - 1 · d d x 3 x + 2 1 - x x + 2 - x 3 3 x + 2 0 - 1 x + 2 + 1 - x 1 + 0 ( 1 - x ) 2 ( x + 2 ) 2 = 3 x 2 3 x + 2 + x 3 ( 3 · 1 + 0 ) 2 3 x + 2 1 - x x + 2 - - 2 x - 1 x 3 3 x + 2 ( 1 - x ) 2 ( x + 2 ) 2 ∴ d d x x 3 3 x + 2 ( 1 - x ) ( x + 2 ) = 1 - x x + 2 3 x 2 3 x + 2 + 3 x 3 2 3 x + 2 - - 2 x - 1 x 3 3 x + 2 ( 1 - x ) 2 ( x + 2 ) 2

Find d y d x of the function y = ( x - 3 ) x 2 + 4 3 x 2 + 4 x + 5 . Find $\frac{d y}{d x}$ of each of the following functions.Find $\frac{d y}{d x}$ of each of the following functions.Find $\frac{d y}{d x}$ of each of the following functions.

Given, y = ( x - 3 ) x 2 + 4 3 x 2 + 4 x + 5 Applying log on both sides, log y = log ( x - 3 ) x 2 + 4 3 x 2 + 4 x + 5 1 2 = 1 2 log ( x - 3 ) x 2 + 4 3 x 2 + 4 x + 5 = 1 2 log ( x - 3 ) + log x 2 + 4 - log 3 x 2 + 4 x + 5 ) Differentiating w.r.t x , we get 1 y · d y d x = 1 2 1 x - 3 + 2 x x 2 + 4 - 6 x + 4 3 x 2 + 4 x + 5 ∴ d y d x = y 2 1 x - 3 + 2 x x 2 + 4 - 6 x + 4 3 x 2 + 4 x + 5