To determine

To find: The first the derivative by finding first few derivatives and then observing the pattern that occurs

Answer

The derivative of y=D103cos2x is 2103sin2x

Explanation

Given:

The function is y=D103cos2x.

Formula used:

Chain Rule:

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

F′(x)=g′(h(x))⋅h′(x) (1)

Calculation:

Obtain the derivative of f(x)=cos2x

f1(x)=ddx(cos2x)=−sin2xddx(2x)=−2sin2x

Similarly obtain the second, third, fourth and fifth derivative of the function as follows.

f2(x)=−2(2)cos2xf3(x)=2(2)(2)(sin2x)f4(x)=2(2)(2)(2)cos2xf5(x)=−2(2)(2)(2)(2)sin2x

Observe that the derivatives that appear in even number are cos2x and that appear in odd number are sin2x, therefore the 103^rd derivative will be of sin2x.

The number of 2’s is exactly equal to the derivative number, hence the derivative will be same as 2n, where n denotes the nth derivative.

The sign of the derivative can be defined with an modulo function, that is if the derivative can be divided by the number 4 then it will have the form cos2x and if it is not divided by 4, then a modulo function can be defined as nmod4=3.

Substitute n=103 in the equation nmod4=3 as,

103mod4=3103−43=993

Notice that it is perfectly divisible by the number and hence the sign is positive.

Hence, the derivative y=D103cos2x can be expressed as 2nsin2x where n=103.

Therefore, the derivative of y=D103cos2x is 2103sin2x.