did i use the chain rule correctly?

y=(8x^4-5x^2+1)^4
d(f(x)/dx =d/dx ((8x^4-5x^2+1)^4)
=4*(8x^4-5x^2+1)^3*d/dx(8x^4-5x^2+1)
=4*(8*d/dx(x^4)-5*d/dx(x^2))*(8x^4-5x^2+1)^3
=4*(8*4x^3-5*2x)*(8x^4-5x^2+1)^3
=4*(32x^3-10x)*(8x^4-5x^2+1)^3

Yes.

(8x^4-5x^2+1)^4

=4(8x^4 - 5x^2 + 1)*d/dx( 8x^4-5x^2+1)

= 4(8x^4 - 5x^2 + 1)^3*(32x^3-10x^2)

All you do is differentiate the outter exponent first. then multiply that by the derivative of the inside parenthesis

Yes, you have correctly applied the chain rule to differentiate the function y = (8x^4 - 5x^2 + 1)^4.

To differentiate the function using the chain rule, you have followed the correct steps:

1. Start with the function y = (8x^4 - 5x^2 + 1)^4.
2. Apply the chain rule: d(f(x)/dx = d/dx((8x^4 - 5x^2 + 1)^4).
3. Multiply by the derivative of the function inside the parentheses: 4 * (8x^4 - 5x^2 + 1)^3 * d/dx(8x^4 - 5x^2 + 1).
4. Apply the derivative to each term inside the parentheses: 4 * (8 * d/dx(x^4) - 5 * d/dx(x^2)) * (8x^4 - 5x^2 + 1)^3.
5. Simplify the derivatives: 4 * (8 * 4x^3 - 5 * 2x) * (8x^4 - 5x^2 + 1)^3.
6. Further simplify: 4 * (32x^3 - 10x) * (8x^4 - 5x^2 + 1)^3.

So, yes, you have correctly used the chain rule to differentiate the given function.