How do I find the derivative of?

Question

f(z)=(root of z)(z^4-2z)

Answer 1

I don't know if there's enough information to answer this question as you've stated it. My thought is that - depending on what information is missing - your answer can be expressed in one of 3 ways:

In mathematics, a root (or a zero) of a function f is a member x of the domain of f such that f(x) vanishes at x, that is:

x : f(x) = 0

Basically, you're looking for the value of x such that f(x) = 0. However, this value of x will be a constant, not a variable, because the roots of a function never change.

With respect to the equation provided, this implies two things: first, that z must be a function, and second, that the root of z (let's cal this R(z)) is a constant.

In this interpretation:

f(z) = R(z)*(z^4 - 2z)
R(z) = constant
z = z(x) (z is a function of an aribtrary variable, x)

Thus, using the chain rule the derivative is found to be:

f ' (z) = R(z) * (4*z^3*dz/dx - 2*dz/dx)

(note that R(z) has no effect on the differentiation because it is a constant)


2ND INTERPRETATION:

Now, if you meant that

f(z)=(root of f(z))(z^4-2z)

then this greatly simplifies the result. The root of f(z) is a constant, just like before, only now z is an independant variable instead of a function. In this case the solution is simply:

f ' (z) = R(f) * (4*z^3 - 2)
(where R(f) is the root, z, such that f(z)=0)


3RD INTERPRETATION

"Root" can also refer to a fractional exponent in mathematics. For instance, x^(1/2) is the square root of x, x^(1/3) is the cube root of x, and so on. Since the fraction of the exponent is not given, we'll just call it 1/n:

f(z) = z^(1/n) * (z^4 - 2z)

Multiply it all out:

f(z) = z^((4n+1)/n) - 2*z^((n+1)/n)

Differentiate using the power rule:

f ' (z) = ((4n+1)/n) * z^((3n+1)/n) - 2*z^(1/n)

Finally, assuming you meant the "square root of z," then n = 2, and the answer becomes:

f ' (z) = ((4*2+1)/2) * z^((3*2+1)/2) - 2*z^(1/2)

f ' (z) = (9/2)*z^(7/2) - 2*z^(1/2)

f ' (z) = (4.5) * z^(3.5) - 2*z^(.5)

Hope that helps!

Answer 2

Finding the derivative is easy f you know the short cut. Multiply the exponent to the coeffient of the number. Then subtract the expoent by 1. In this case, if you meant (z)^(1/2)*(z)^(4)-2z, (note that raising something to the 1/2 is the same as the square root). The derivatie is (1/2)z^(-1/2)*4z^(3)-2.

Answer 3

z^.5 * (z^4 - 2z) equals z^4.5 - 2 Z^1.5

From then, it just a matter of using the standard definition for derivative, which is

d/dx x^n = n x^(n-1)

so the derivative will be

4.5 z^3.5 - 3 z^.5

(where z^.5 is "square root of z")

Answer 4

root of z = z^½, so you can rewrite the equation as:

z^4½ - 2z^1½

The derivative will then be:

4½*z^3½ - 3z^½

Answer 5

f(z)=√x *(z^4-2z)=z^4.5-2z^1.5
f'(z)=4.5z^3.5-3z^.5=4.5z^3.5*√z-3√z