To determine

To find: The derivative of y=sin1+x2.

Answer

The derivative of y=sin1+x2 is xcos(1+x2)1+x2.

Explanation

Derivative Rules:

The Chain Rule:

If g is differentiable at x and f is differentiable at g(x), then the composite function F=f∘g defined by F(x)=f(g(x)) is differentiable at x and F′ is given by the product

F′(x)=f′(g(x))⋅g′(x) (1)

(1).Power Rule: ddx(xn)=nxn−1

(2) Sum Rule: ddx[f(x)+g(x)]=ddx[f(x)]+ddx[g(x)]

Calculation:

Obtain the derivative of y.

y′=ddx(sin1+x2)

Apply chain rule and simplify as,

y′=(cos1+x2)ddx(1+x2)

Apply chain rule, power rule and simplify as,

y′=(cos1+x2)ddx(1+x2)12=(cos1+x2)12((1+x2)12−1)ddx(1+x2)=(cos1+x2)12(1(1+x2))ddx(1+x2)

Apply sum rule and simplify as,

y′=(cos1+x2)12(1(1+x2))[ddx(1)+ddx(x2)]=(cos1+x2)(12(1+x2))[0+2x2−1]=(cos1+x2)(2x2(1+x2))=(cos1+x2)(x(1+x2))

Therefore, the derivative of y=sin1+x2 is xcos(1+x2)1+x2.