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.