Ralf,

I like this problem. I'm going to do one like it for you, but with three parts.

Find the derivative of the function
1 -. y = 3 - 5x⁴
2 -. y = 3 - (5x)⁴
3 -. y = (3 - 5x )⁴

1. This is an application of the first properties of differentiation. the derivative of a constant is zero, the derivative of x^r is rx^r-1 and if k is a constant and f(x) is a differentiable function then the derivative of k f(x) is k f '(x). Thus the derivative of y = 3 - 5x⁴ is dy/dx = -5 (4x³) = -20 x³.
   
   
2. For the second problem you need the properties I stated before plus the chain rule. One special case of the chain rule is that if f(x) is a differentiable function and k is a constant then the derivative of (f(x))^k is k (f(x))^k-1 f '(x). Hence the derivative of y = 3 - (5x)⁴ is dy/dx = -4 (5x)³ × 5 = -20 (5x)³.
   
   
3. This is an application of the chain rule again. Since the derivative of 3 - 5x is -5 the derivative of y = (3 - 5x )⁴ is dy/dx = 4(3 - 5x)³ × (-5) = -20 (3 - 5x)³

I hope this helps,
Harley