What is the derivative of the fifth root of x^3?
if y = (x^3)^(1/5)
= x^(3/5)
we would have dy/dx = (3/5)x^(3/5 - 1) = (3/5) x^(-2/5)
fifth root of x^3 = x(3/5)
derivative x^(3/5) = 3/5 x^(-2/5) = 3/5 *[5th root (x^-2)]
Thanks, i get it now
y = (x^3)^(1/5)
Using the chain rule,
y' = 1/5 (x^3)(-4/5) * 3x^2 = 3/5 x^(-12/5 + 2) = 3/5 x^(-2/5)
To find the derivative of the fifth root of x cubed, we can use the chain rule.
Let's start by expressing the function in a different form that is easier to differentiate. The fifth root of x cubed can also be written as (x^3)^(1/5).
Now, let's find the derivative step by step.
Step 1: Apply the power rule to differentiate the function x^3.
The power rule states that if we have a function of the form f(x) = x^n, then its derivative is given by f'(x) = n*x^(n-1).
In this case, n = 3. So, the derivative of x^3 with respect to x is 3*x^(3-1) = 3x^2.
Step 2: Apply the chain rule.
The chain rule states that if we have a composite function of the form f(g(x)), where f(u) and g(x) are differentiable functions, then the derivative of f(g(x)) with respect to x is given by f'(g(x)) * g'(x).
In our case, f(u) = u^(1/5) and g(x) = x^3. Let's find the derivatives of f(u) and g(x) separately.
The derivative of f(u) = u^(1/5) can be found by applying the power rule. Let m = 1/5.
So, the derivative of f(u) = u^(1/5) with respect to u is (1/5) * u^((1/5)-1) = (1/5) * u^(-4/5).
The derivative of g(x) = x^3, as we calculated earlier, is 3x^2.
Finally, we can apply the chain rule by multiplying the derivatives f'(u) and g'(x):
f'(u) * g'(x) = (1/5) * u^(-4/5) * 3x^2.
Since u = x^3, we can replace u with x^3:
= (1/5) * (x^3)^(-4/5) * 3x^2.
= (1/5) * x^(-12/5) * 3x^2.
Simplifying further:
= (3/5) * x^(2-(12/5)).
= (3/5) * x^(-2/5).
Therefore, the derivative of the fifth root of x^3 is (3/5) * x^(-2/5).