Question

Решить производную y=x^3(6-3x+x^4)

Answer 1

y=x^3(6-3x+x^4)

Применим формулу: (UV)'= U'V + UV'

Решение:

y'= 3x²(6-3x+x^4) + x³(-3 +4x³) = 18x² -9x³ +3x⁶ -3x³ +4x⁶ = 7x⁶-12x³ +18x²

Answer 2

To find the derivative of the given function $y = x^3(6 - 3x + x^4)$ with respect to $x$, we can use the product rule and chain rule of differentiation. The product rule states that if we have a function of the form $y = u \cdot v$, where $u$ and $v$ are both functions of $x$, then the derivative of $y$ with respect to $x$ is given by:

$\frac{{dy}}{{dx}} = u'v + uv',$

where $u'$ and $v'$ are the derivatives of $u$ and $v$ with respect to $x$, respectively.

In our case, let's consider $u = x^3$ and $v = (6 - 3x + x^4)$. Now, let's find the derivatives of $u$ and $v$ and apply the product rule:

$u' = \frac{{d}}{{dx}}(x^3) = 3x^2,$ $v' = \frac{{d}}{{dx}}(6 - 3x + x^4) = -3 + 4x^3.$

Now, using the product rule:

$\frac{{dy}}{{dx}} = u'v + uv' = (3x^2)(6 - 3x + x^4) + (x^3)(-3 + 4x^3).$

Now, simplify the expression:

$\frac{{dy}}{{dx}} = 18x^2 - 9x^3 + 3x^6 - 3x^4.$

So, the derivative of $y$ with respect to $x$ is $18x^2 - 9x^3 + 3x^6 - 3x^4$.