we have (iv) Let y = e cos x . Using chain rule, we have ( sin ) (sin ) x e ⋅− = − EXERCISE . Differentiate the following w.r.t. x : .
x e . . x e . sin (tan – e – x ) .
log (log x ), x > . cos , > . cos (log x + e x ), x > . .
Logarithmic Differentiation In this section, we will learn to differentiate certain special class of functions given in the form y = f ( x ) = [ u ( x )] v ( x ) By taking logarithm (to base e ) the above may be rewritten as log y = v ( x ) log [ u ( x )] Using chain rule we may differentiate this to get v x y dx u x . u ′ ( x ) + v ′ ( x ) . log [ u ( x )] which implies that [ ] ( ) log v x u x v x u x u x ⋅′ + ′ The main point to be noted in this method is that f ( x ) and u ( x ) must always be positive as otherwise their logarithms are not defined. This process of differentiation is known as logarithms differentiation and is illustrated by the following examples: Example Differentiate ) ( ) w.r.t.
x . Solution Let ) ( ) ( ) Taking logarithm on both sides, we have log y = [log ( x – ) + log ( x + ) – log ( x + x + )] Now, differentiating both sides w.r.t. x , we get dy y dx ( ) or ) )( ) ) Example Differentiate a x w.r.t. x , where a is a positive constant.
Solution Let y = a x .