Differentiate f(x)=2x^(2/3)(3-4x^(1/3))

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In summary, the degree of the polynomial is 2 and the leading coefficient is 2. To find the critical points of the function, set the derivative equal to 0 and solve for x, giving us a critical point of (1/2, 0). The domain of the function is all real numbers except for x = 0. To find the x-intercepts, set the function equal to 0 and solve for x, giving us x = 0 and x = 3/4. The y-intercept is (0,0).
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shanshan
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Homework Statement


differentiate f(x) = [2x^(2/3)][3-4x^(1/3)]


Homework Equations





The Attempt at a Solution


f'(x) = (4/3)(x^(-1/3))(3-4x^(1/3)) + (1/3)(-4)(x^(-2/3))
= 4x^(-1/3)-(16/3)-((4/3)(x^(-2/3)))
= (4/x^(1/3))-(16/3)-(4/3(x^(-2/3)))
I'm pretty sure I've made a mistake, but I can't find it.
 
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  • #2
Remember, the product rule is defined as [itex]f' = g'h + gh'[/itex]. When you used the product rule, you forgot to multiply the second term by [itex]2x^{2/3}[/itex].
 

FAQ: Differentiate f(x)=2x^(2/3)(3-4x^(1/3))

What is the degree of the polynomial?

The degree of the polynomial is 2. This is because the highest exponent in the polynomial is 2.

What is the leading coefficient?

The leading coefficient is 2. This is because it is the coefficient of the term with the highest degree, which is x^(2/3).

How do I find the critical points of the function?

To find the critical points, set the derivative of the function equal to 0 and solve for x. In this case, the derivative is f'(x) = 4x^(1/3) - 2x^(-1/3). Solving for x, we get x = 1/2. Therefore, the critical point of the function is (1/2, 0).

What is the domain of the function?

The domain of the function is all real numbers except for x = 0. This is because the function is undefined at x = 0 due to the presence of x^(-1/3).

How do I find the x-intercepts and y-intercepts of the function?

To find the x-intercepts, set the function equal to 0 and solve for x. In this case, we get x = 0 and x = 3/4. Therefore, the x-intercepts of the function are (0,0) and (3/4,0). To find the y-intercept, plug in x = 0 into the function and we get y = 0. Therefore, the y-intercept of the function is (0,0).

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