Question

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

Answer 1

The 57th derivative of y = cos(7x) is given by 7^57 * cos(7x + (57*pi/2))

Step-by-step explanation:

To find the 57th derivative of y = cos(7x), we can use the formula for the nth derivative of a cosine function:

d^n(cos(ax)) = a^n * cos(ax + (n*pi/2))

In this case, a = 7, so we can substitute a = 7 into the formula:

d^57(cos(7x)) = 7^57 * cos(7x + (57*pi/2))

Answer 2

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)