Question

Find the given derivative by finding the first few derivatives and observing the pattern that occurs. . . d^114 /dx^114 of (sinx)

Answer 1

So in your given pattern, you need to find first the derivatives and observe the patter that occurs in the given functions. So with this kind of pattern, every fourth one is the same; that makes the 114th derivative is the same as the second derivative. It is known since 114/4 has a remainder of two

Answer 2

To find

$\frac{d^{114}}{dx^{114}}(sinx)$

We need to find some few derivatives of $sinx$, say the first eight derivatives and then observe some pattern.

$\frac{dy}{dx}(sinx)=cosx$

$\frac{d^{2}}{dx^{2}}(sinx)=-sinx$

$\frac{d^{3}}{dx^{3}}(sinx)=-cosx$

$\frac{d^{4}}{dx^{4}}(sinx)=sinx$

$\frac{d^{5}}{dx^{5}}(sinx)=cosx$

$\frac{d^{6}}{dx^{6}}(sinx)=-sinx$

$\frac{d^{7}}{dx^{7}}(sinx)=-cosx$

$\frac{d^{8}}{dx^{8}}(sinx)=sinx$

We can recognize the following patterns in the order of the derivativatives;

1. The derivative of $sinx$ to the orders, $1,5,9,13,...,4n-3$ is $cosx$

2. The derivative of $sinx$ to the orders, $2,6,10,14,...,4n-2$ is $-sinx$

3. The derivative of $sinx$ to the orders, $3,7,11,15,...,4n-1$ is $-cosx$

The order, 114 belongs to the second pattern. Therefore,

$\frac{d^{114}}{dx^{114}}(sinx)=-sinx$