To determine

To find: The derivative of d99dx99(sinx).

Answer

The derivative of d99dx99(sinx) is −cosx_.

Explanation

Let f(x)=sinx, then d99dx99(f(x))=d99dx99(sinx).

The first derivative of f(x) is computed as follows,

ddx(f(x))=ddx(sinx)=cosx

Thus, the first derivative of f(x) is ddx(f(x))=cosx (1)

The second derivative of f(x) is computed as follows,

d2dx2(f(x))=ddx(ddx(f(x)))=ddx(cosx)            [by(1)]=−sinx

Thus, the second derivative of f(x) is d2dx2(f(x))=−sinx (2)

The third derivative of f(x) is computed as follows,

d3dx3(f(x))=ddx(d2dx2(f(x)))=ddx(−sinx)            [by(2)]=−ddx(sinx)=−cosx

Thus, the third derivative of f(x) is d3dx3(f(x))=−cosx (3)

The fourth derivative of f(x) is computed as follows,

d4dx4(f(x))=ddxd3dx3(f(x))=ddx(−cosx)         [by(3)]=−(−sinx)=sinx

Thus, the fourth derivative of f(x) is d4dx4(f(x))=sinx (4)

Note that, every fourth derivative is the original function f(x).

Therefore, the given derivative d99dx99(sinx) can be written as follows,

d99dx99(f(x))=d4(24)+3dx4(24)+3(f(x))    [Q99=4(24)+3]=d3dx3(d96dx96(f(x)))=d3dx3(sinx)         [by (4)]=−cosx               [by (3)]

Therefore, the derivative of d99dx99(sinx) is d99dx99(sinx)=−cosx_.