How do I find the derivative of y=(8x^4-5x^2+1)^4
To find the derivative of a function, you can follow these steps:
1. Expand the given function using the binomial theorem, which states that (a + b)^n = Σ C(n, r) * a^(n-r) * b^r, where C(n, r) is the binomial coefficient and Σ indicates summation. In this case, a = 8x^4 - 5x^2 + 1 and b = 0.
The fourth power of a binomial can be expanded as:
y = (8x^4 - 5x^2 + 1)^4
= C(4, 0) * (8x^4)^4 * (0)^0 + C(4, 1) * (8x^4)^3 * (-5x^2)^1 * (0)^0 + C(4, 2) * (8x^4)^2 * (-5x^2)^2 * (0)^0 + C(4, 3) * (8x^4)^1 * (-5x^2)^3 * (0)^0 + C(4, 4) * (8x^4)^0 * (-5x^2)^4 * (0)^0
= (8x^4)^4 + 4 * (8x^4)^3 * (-5x^2) + 6 * (8x^4)^2 * (-5x^2)^2 + 4 * (8x^4)^1 * (-5x^2)^3 + (-5x^2)^4
= 4096x^16 - 102400x^14 + 998400x^12 - 512000x^10 + 156250x^8
2. Differentiate each term of the expanded function using the power rule. The power rule states that if you have a term of the form ax^n, where a is a constant and n is a positive integer, the derivative is given by d(ax^n)/dx = nax^(n-1).
Therefore, to differentiate the expanded function, we differentiate each term with respect to x:
dy/dx = d(4096x^16)/dx - d(102400x^14)/dx + d(998400x^12)/dx - d(512000x^10)/dx + d(156250x^8)/dx
= 16 * 4096x^(16-1) - 14 * 102400x^(14-1) + 12 * 998400x^(12-1) - 10 * 512000x^(10-1) + 8 * 156250x^(8-1)
= 65536x^15 - 1433600x^13 + 11980800x^11 - 5120000x^9 + 1250000x^7
So, the derivative of y = (8x^4 - 5x^2 + 1)^4 is dy/dx = 65536x^15 - 1433600x^13 + 11980800x^11 - 5120000x^9 + 1250000x^7.