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 103rd 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.