Find the 57th derivative of y = cos (7x).

y = cos (7x)

First derivative = - 7sin (7x)

Second derivative = - (7^2) cos (7x)

3rd derivative = (7^3) sin (7x)

4th derivative = (7^4) cos (7x)

5th derivative = - (7^5) sin (7x)

6th derivative = - (7^6) cos (7x)

nth derivative?

Analysis and conclusion

If n is multiple of 4 or a number before a multiple of 8 the sign is positive

If n is one or two numbers after a multiple of 4 the sign is negative.

7 is raised to exponent n

if n is odd the function is sin (7x), if n is even the function is cos (7x)

n = 57 = > odd, and it is a number after 56 which is a multiple of 4 = > negative sign

Therefore, the 57th derivative = - (7^57) sin (7x)