Question

x^3sinx/x^2+5
differentiate with respect to x​

Answer 1

Answer:

[ ( x² + 5 ) ( x³ . cos x + 3 x² . sin x + ( 2 x⁴ . sin x ) ] / [ ( x² + 5 ) ]²

Step-by-step explanation:

Given :

y = ( x³ sin x ) / ( x² + 5 )

Using quotient rule :

= > d / d x [ f ( x ) / g ( x ) ] = [ g ( x ) . f' ( x ) - f ( x ) . g' ( x ) ] / [ g ( x ) ]²

Diff. w.r.t. x :

y' = [ ( x² + 5 ) ( x³ sin x )' + ( x³ sin x ) ( x² + 5 )' ] / [ ( x² + 5 ) ]²

y' = [ ( x² + 5 ) ( x³ ( sin x )' + sin x ( x³ )' ) + ( x³ sin x ) ( x² + 5 )' ] / [ ( x² + 5 ) ]²

y' = [ ( x² + 5 ) ( x³ ( sin x )' + sin x ( x³ )' + ( x³ sin x ) ( 2 x + 0 ) ] / [ ( x² + 5 ) ]²

y' = [ ( x² + 5 ) ( x³ ( sin x )' + sin x ( x³ )' + ( x³ sin x ) ( 2 x ) ] / [ ( x² + 5 ) ]²

y' = [ ( x² + 5 ) ( x³ . cos x + 3 x² . sin x + ( x³ sin x ) ( 2 x ) ] / [ ( x² + 5 ) ]²

y' = [ ( x² + 5 ) ( x³ . cos x + 3 x² . sin x + ( 2 x⁴ . sin x ) ] / [ ( x² + 5 ) ]²

Hence we get required answer!