y . log x ) dy dx = – x x ( + log x ) – y . x y – – y x log y Therefore [ . ( log )] .
y x x y EXERCISE . Differentiate the functions given in Exercises to w.r.t. x . .
cos x . cos x . cos x . ) ( ) ) ( ) ( ) .
(log x ) cos x . x x – sin x . ( x + ) . ( x + ) .
( x + ) . . (log x ) x + x log x . (sin x ) x + sin – .
x sin x + (sin x ) cos x . . ( x cos x ) x + ( sin ) x Find dy dx of the functions given in Exercises to . .
x y + y x = . y x = x y . (cos x ) y = (cos y ) x . xy = e ( x – y ) .
Find the derivative of the function given by f ( x ) = ( + x ) ( + x ) ( + x ) ( + x ) and hence find f ′ ( ). . Differentiate ( x – x + ) ( x + x + ) 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? . If u , v and w are functions of x , then show that dx ( u . v .
w ) = du dx v . w + u . dv dx . w + u .
v dw in two ways - first by repeated application of product rule, second by logarithmic differentiation. . Derivatives of Functions in Parametric Forms Sometimes the relation between two variables is neither explicit