I need help finding the derivative of this function:
f(x)=(X^2+2)^4/5
Thanks in advance! Any help would be greatly appreciated!
just use the chain rule
f'(x) = (4/5) (x^2+2)^(-1/5) (2x)
= ...
You break it into parts to turn it into a simpler equation so:
set u= x^2 + 2
then your equation becomes:
f(u)= u^4/5
and f'(x) = f'(u)* u'
f'(u) = 4/5 * u^(4/5-1) = 4/5 * u^-1/5
Now take the derivative of u to get
u' = 2x
Then substitute to get in terms of x
f'(x) =4/5 * (x^2 + 2)^-1/5 * 2x
Finding the derivative of a function involves using the rules of calculus, specifically the power rule and the chain rule.
Let's start by applying the power rule, which states that if you 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, we have f(x) = (x^2 + 2)^(4/5). To find the derivative, we will need to apply the chain rule since we have a function raised to a fractional power.
The chain rule states that if we have a function h(x) = g(f(x)), where g(x) and f(x) are differentiable, then the derivative of h(x) with respect to x is given by h'(x) = g'(f(x)) * f'(x).
First, let's find the derivative of the inner function, f(x) = x^2 + 2. The derivative of f(x) with respect to x is f'(x) = 2x.
Now, let's find the derivative of the outer function, g(x) = x^(4/5). Applying the power rule, g'(x) = (4/5)*x^((4/5)-1) = (4/5)*x^(-1/5) = (4/5)/x^(1/5) = 4/5x^(1/5).
Finally, we can apply the chain rule to find the derivative of f(x) = (x^2 + 2)^(4/5):
f'(x) = g'(f(x)) * f'(x)
= (4/5) * (x^2 + 2)^(-1/5) * 2x
Therefore, the derivative of f(x) = (x^2 + 2)^(4/5) is f'(x) = (4/5) * (x^2 + 2)^(-1/5) * 2x.
I hope this explanation helps! Let me know if there's anything else I can explain.