Differeniate
V^3-2v√v
I used the quotient rule
(v)(v^3-2v√v)' -(v^3-2v√v)(v) /(v)'
The answer is 2v-1-1
I got this far but then I don't know what to do further
2v^3-2v^1/2+2v√v/(v)^2
f(v) = v^3 - 2v^(3/2)
f ' (x) = 3v^2 - 3v^(1/2)
= 3v^2 - 3/√v
Why would you think of using the quotient rule ? There is no division .
Idon't why my phone didn't put the end part the problem is v^3-2v√v/v
Idon't why my phone didn't put the end part the problem is v^3-2v√v/v
To further simplify the expression, you need to apply the power rule and distribute the denominator. Let's break it down step by step.
Step 1: Apply the quotient rule and find the derivative of the given expression:
(v)(v^3 - 2v√v)' - (v^3 - 2v√v)(v)' / (v)'.
First, let's find the derivative of v^3 - 2v√v using the power rule:
(v^3 - 2v√v)' = 3v^2 - (2)(1/2)v^(1/2 + 1).
Simplifying further:
(v^3 - 2v√v)' = 3v^2 - v^(3/2).
Next, let's find the derivative of v:
(v)' = 1.
Now, we can substitute these derivatives back into the quotient rule expression:
(3v^2 - v^(3/2))v - (v^3 - 2v√v)(1) / 1.
Simplifying this expression further:
3v^3 - v(2v√v) - (v^3 - 2v√v).
Multiplying through:
3v^3 - 2v^2√v - v^3 + 2v√v.
Combining like terms:
2v^3 - 2v^2√v + 2v√v.
Step 2: Factor out the common term from the expression:
2v(v^2 - v√v + √v).
Step 3: Simplify the expression inside the parentheses.
Let x = √v. Substituting x back into the expression gives us:
2v(v^2 - vx + x).
Now, let's use the quadratic formula to simplify the expression inside the parentheses:
v = (-(-vx) ± √((-vx)^2 - 4(x)(1))) / (2(1)).
v = (vx ± √(v^2x^2 - 4x)) / 2.
v = (vx ± √(v^2x^2 - 4x)) / 2.
v = (vx ± √(v^2(x^2 - 4))) / 2.
v = (vx ± √(v^2)√(x^2 - 4)) / 2.
v = (vx ± v√(x^2 - 4)) / 2.
v = v(x ± √(x^2 - 4)) / 2.
v = (x ± √(x^2 - 4)) / 2.
Substituting x back in, we get:
v = (√v ± √(√v^2 - 4)) / 2.
Step 4: Substitute the value of v back into the factored expression:
2v(v^2 - v√v + √v) = 2(√v ± √(√v^2 - 4))(√v^2 - v√v + √v).
And that's the final simplified expression after differentiating and simplifying using the power and quotient rules.