In addition, can you walk me through how to get the derivatives for these 2 statements, too?
a) y = x^5/3 - 5x^2/3
b) y = (the cubed root of the quantity) [(x^2 - 1)^2]
Hi there. I need to find the first derivative of this statement.
y=x(x+2)^3
I tried the chain rule, but I think I messed up because I got y'=3(x+2)^5.
The reason I know it's wrong is because I'm trying to sketch the curve and my points are not where they should be.
Help?
Use primarily the product rule
y = x(x+2)^3
y' = (1)(x+2)^3 + x(3)(x+2)^2
=(x+2)^2(1+3x)
for your second post
a) y'=(5/3)x^(2/3) - (10/3)x^(-1/3)
b) y = [(x^2-1)^2]^(1/3)
= (x^2-1)^(2/3)
then y'=(2/3)(x^2-1)^(-2/3)(2x) etc
Thanks so much!
No problem! I'll walk you through how to find the derivatives of the given statements.
a) To find the derivative of y = x^(5/3) - 5x^(2/3), you can use the power rule for differentiation. The power rule states that if you have a function of the form f(x) = x^n, then the derivative of that function is f'(x) = n*x^(n-1).
Applying the power rule to each term:
For the first term, we have y = x^(5/3), so the derivative is y' = (5/3)*x^(5/3 - 1) = (5/3)*x^(2/3).
For the second term, we have y = -5x^(2/3), so the derivative is y' = -5*(2/3)*x^(2/3 - 1) = -10/3*x^(-1/3).
Therefore, the derivative of y = x^(5/3) - 5x^(2/3) is y' = (5/3)*x^(2/3) - (10/3)*x^(-1/3).
b) To find the derivative of y = [(x^2 - 1)^2]^(1/3), we can apply the chain rule, along with the power rule.
Let's simplify the expression first.
y = [(x^2 - 1)^2]^(1/3) = (x^2 - 1)^(2/3).
Now, to find the derivative, we use the power rule:
y' = (2/3)*(x^2 - 1)^(2/3 - 1)*(2x) = (2/3)*(x^2 - 1)^(-1/3)*(2x).
That's it! The derivative of y = [(x^2 - 1)^2]^(1/3) is y' = (2/3)*(x^2 - 1)^(-1/3)*(2x).
I hope this clears things up! Let me know if you have any more questions.