Question

Дифференциал функции y = (ax^2-b)^3 равен 1. 4ax(ax^2-b)dx 2. 2ax(ax-b)^2 dx 3. 2ax(ax^2-b)^2 dx 4. (ax^2 -b)^3 dx 4. 2ax(ax^2-b)dx

Answer 1

$y=(ax^2-b)^3\\\\dy=y'\cdot dx\\\\dy=3\cdot (ax^2-b)^2\cdot 2ax\cdot dx=6ax\cdot (ax^2-b)^2\cdot dx$

P.S. Если  $y=(ax^2-b)^2$  ,  то $dy=4ax(ax^2-b)\, dx$  .

Answer 2

To find the differential of the function $y = (ax^2 - b)^3$, we need to take the derivative with respect to $x$ and then multiply it by $dx$. Let's proceed with the differentiation step:

$y = (ax^2 - b)^3$

To find $\frac{dy}{dx}$, we'll use the chain rule:

$\frac{dy}{dx} = 3(ax^2 - b)^2 \cdot \frac{d}{dx}(ax^2 - b)$

Now, let's find $\frac{d}{dx}(ax^2 - b)$:

$\frac{d}{dx}(ax^2 - b) = 2ax$

Now, we can substitute this back into the original expression:

$\frac{dy}{dx} = 3(ax^2 - b)^2 \cdot 2ax$

Simplifying:

$\frac{dy}{dx} = 6ax(ax^2 - b)^2$

So, the correct option for the differential of the function $y = (ax^2 - b)^3$ is:

$6ax(ax^2 - b)^2 \, dx$

Option 3 is the correct answer:

3. $2ax(ax^2 - b)^2 \, dx$