#### 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 *n*th 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.