Question

Question #eb80b

Answer 1

`(x(2-9x^3))/(1-3x^3)^(2/3).`

Explanation:

Suppose that, `y=x^2(1-3x^3)^(1/3).`

Applying the Product Rule, we get,

`dy/dx=x^2*d/dx(1-3x^3)^(1/3)+(1-3x^3)^(1/3)*d/dx(x^2)....(star).`

Next, we use the Power Rule, and, the Chain Rule, to get,

`d/dx(1-3x^3)^(1/3)=1/3*(1-3x^3)^(1/3-1)*d/dx(1-3x^3),`

`=1/3*(1-3x^3)^(-2/3)*(0-3*3x^2),`

`rArr d/dx(1-3x^3)^(1/3)=(-3x^2)/(1-3x^3)^(2/3).......(star_1).`

Using `(star_1)` in `(star),` we get,

`dy/dx=x^2{(-3x^2)/(1-3x^3)^(2/3)}+(1-3x^3)^(1/3)*2x,`

`=2x(1-3x^3)^(1/3)-(3x^4)/(1-3x^3)^(2/3),`

`={2x(1-3x^3)^(1/3)(1-3x^3)^(2/3)-3x^4}/((1-3x^3)^(2/3),`

`={2x(1-3x^3)-3x^4}/(1-3x^3)^(2/3).`

`rArr dy/dx=(x(2-9x^3))/(1-3x^3)^(2/3).`

Enjoy Maths.!

Answer 2

`frac(d)(dx)(x^(2) (1 - 3 x^(3))^(frac(1)(3))) = - 3 x^(4) (1 - 3 x^(3))^(- frac(2)(3)) + 2 x (1 - 3 x^(3))^(frac(1)(3))`

Explanation:

We have: `x^(2) (1 - 3 x^(3))^(frac(1)(3))`

We need to find the derivative of the function `f(x) = x^(2) (1 - 3 x^(3))^(frac(1)(3))`.

Let's begin differentiating `f(x)` by applying the product rule:

`Rightarrow f'(x) = x^(2) cdot frac(d)(dx)((1 - 3 x^(3))^(frac(1)(3))) + (1 - 3 x^(3))^(frac(1)(3)) cdot frac(d)(dx)(x^(2))`

Then, let's use the power rule to further differentiate:

`Rightarrow f'(x) = x^(2) cdot frac(d)(dx)((1 - 3 x^(3))^(frac(1)(3))) + (1 - 3 x^(3))^(frac(1)(3)) cdot 2 x`

Now, the final bit of differentiation must be done by using the chain rule.

Suppose that:

`u = 1 - 3 x^(3) Rightarrow u' = frac(d)(dx)(1) - frac(d)(dx)(3 x^(3)) Rightarrow u' = 0 - 3 cdot 3 x^(3 - 1) Rightarrow u' = - 9 x^(2)`

`and`

`v = u^(frac(1)(3)) Rightarrow v' = frac(1)(3) u^(frac(1)(3) - 1) Rightarrow v' = frac(1)(3) u^(- frac(2)(3))`

So:

`Rightarrow f'(x) = x^(2) cdot u' cdot v' + 2 x (1 - 3 x^(3))^(frac(1)(3))`

`Rightarrow f'(x) = x^(2) cdot (- 9 x^(2)) cdot frac(1)(3) u^(- frac(2)(3)) + 2 x (1 - 3 x^(3))^(frac(1)(3))`

`Rightarrow f'(x) = - 9 x^(4) cdot frac(1)(3) (1 - 3 x^(3))^(- frac(2)(3)) + 2 x (1 - 3 x^(3))^(frac(1)(3))`

`therefore f'(x) = - 3 x^(4) (1 - 3 x^(3))^(- frac(2)(3)) + 2 x (1 - 3 x^(3))^(frac(1)(3))`