How do you find the derivative of
F(x) = root5(3 + 5 x + x^3)?
f(x)=u^k
f'= ku^(k-1) du
so here, that does something like this..
I assume root5 means the 1/5 power.
f'= 1/5( )^(-4/5) * (5 + 3x^2)
5/2root(3+5x+x^3) * 5+3x^2=
(5(5+3x^2))/(2root(3+5x+x^2)
Is that the 5th root of (3+5x+x^3)? If so, you can rewrite it as (3+5x+x^3)^(1/5). You have to use the chain rule to find this derivative. First set (3+5x+x^3)=z. You know have z^(1/5). Take the derivative of this. (1/5)z^(1/5-1)=(1/5)z^(-4/5). Now substitute (3+5x+x^3) back in for z. (1/5)(3+5x+x^3)^(-4/5). Take the derivative of the inside, which is (3+5x+x^3). You get (0+5+3x^2)=(5+3x^2). Multiple this with (1/5)(3+5x+x^3)^(-4/5). You get (1/5)(5+3x^2)(3+5x+x^3)^(-4/5).
Anonymous is wrong, don't use that. bobpursley and I are correct.
To find the derivative of the function F(x) = √(3 + 5x + x^3), we can make use of the chain rule. The chain rule states that if we have a composite function, such as √(g(x)), then the derivative with respect to x can be found by multiplying the derivative of the outer function (in this case, the square root function) by the derivative of the inner function (in this case, 3 + 5x + x^3).
To apply the chain rule, we first need to find the derivative of the outer function, which is the square root function. The derivative of √x with respect to x is 1/(2√x). Therefore, the derivative of √(3 + 5x + x^3) with respect to x is:
1 / (2√(3 + 5x + x^3)) * (d/dx (3 + 5x + x^3))
Now, we need to find the derivative of the inner function, 3 + 5x + x^3. The derivative of a constant (such as 3) with respect to x is 0. The derivative of 5x with respect to x is 5. The derivative of x^3 with respect to x is 3x^2.
Combining these derivatives, we get:
1 / (2√(3 + 5x + x^3)) * (0 + 5 + 3x^2)
Simplifying further, we get:
(5 + 3x^2) / (2√(3 + 5x + x^3))
Therefore, the derivative of F(x) = √(3 + 5x + x^3) is (5 + 3x^2) / (2√(3 + 5x + x^3)).