Why is it when I try to get a derivative using the power rule I get a different result if I use the Chain rule

For example:
f(x)=5x^2
f'(x)=10x <--answer
(Power Rule f(x)=x^n, f'(x)=nx^(n-1))

f(x)=5x^2
g(x)=z=5x
f(g(x))=f(z)=z^2
f'(g(x))=f'(z)=2z
g'(x)=z'=5
f'(z)*z'=(2z)(5)
=2(5x)(5)
f'(x)=50x <--answer
(Chain Rule f'(g(x))=f'(g(x))*(g'(x))

As you can see there's a big difference.
Am I understanding the chain rule incorrectly?

Your algebra is incorrect.

In the second case, letting z=5x would result in (5x)^2 = 25x^2, whose derivative would be 50x

I don't think you do see Jeff's point. In the first case you're computing the derivative of 5 * x^2, whereas in the second case you're computing the derivative of 25 * x^2, because you made a mistake in your algebra. To repair your work, you would need to set g(x) = sqrt(5)*x.

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