Question

Differentiate and simplify ??

Answer 1

(a) `(df)/(dx)=(4x^11(8x+3))/(2x+1)^5`

(b) `(df)/(dx)=(e^xcos(sqrt(2e^x+1)))/sqrt(2e^x+1)`

Explanation:

(a) As `f(x)=(x^3/(2x+1))^4`

hence using first function of function rule (also called chain rule) and then quotient rule, we get the differential

`(df)/(dx)=4(x^3/(2x+1))^3xxd/dxx^3/(2x+1)`

= `4(x^3/(2x+1))^3xx(3x^2(2x+1)+x^3xx2)/(2x+1)^2`

= `4(x^3/(2x+1))^3xx(8x^3+3x^2)/(2x+1)^2`

= `(4x^9(8x^3+3x^2))/(2x+1)^5`

= `(4x^11(8x+3))/(2x+1)^5`

(b) Again using chain rule, as `f(x)=sin(sqrt(2e^x+1))`

`(df)/(dx)=cos(sqrt(2e^x+1))xxd/dx(sqrt(2e^x+1))`

= `cos(sqrt(2e^x+1))xx1/(2sqrt(2e^x+1))xxd/dx(2e^x+1)`

= `cos(sqrt(2e^x+1))xx1/(2sqrt(2e^x+1))xx2e^x`

= `(e^xcos(sqrt(2e^x+1)))/sqrt(2e^x+1)`