find d^85y/dx85 if y = sin x ??
I want the final rule because the solution will be repeated and therefore there will be a particular rule?
If y = sin x
dy/dx = cos x
d2y/dx2 = -sin x
d3y/dx3 = -cos x
d4y/dx4 = sin x
...
d4ky/dx4k = sin x
d4k+1y/dx4k = cos x
d4k+2y/dx4k = -sin x
d4k+3y/dx4k = -cos x
small typo. Should be dx4k+1 etc.
To find the 85th derivative of y = sin(x), you can use the general rule to differentiate sin(x) repeatedly. The derivative of sin(x) is cyclic with a pattern that repeats every four derivatives. The pattern is as follows:
1st derivative: d/dx(sin(x)) = cos(x)
2nd derivative: d^2/dx^2(sin(x)) = -sin(x)
3rd derivative: d^3/dx^3(sin(x)) = -cos(x)
4th derivative: d^4/dx^4(sin(x)) = sin(x)
Since this pattern repeats every four derivatives, we can simplify the problem by finding the remainder when dividing the exponent 85 by 4.
Dividing 85 by 4, we get a quotient of 21 with a remainder of 1.
Therefore, the 85th derivative of y = sin(x) is equivalent to the 1st derivative of y = sin(x), which is cos(x).
Therefore, d^85y/dx^85 = cos(x).