Question

F(x)=√(2x+3)^4 - Найти производную функции.​

Answer 1

$f(x) = \sqrt{ {(2x + 3)}^{4} } = {(2x + 3)}^{2} \\ \frac{d}{dx} ( {(2x + 3)}^{2} ) = 2(2x + 3) \times 2 = 8x + 12$

Answer 2

To find the derivative of the function $F(x) = \sqrt{(2x+3)^4}$, we can apply the chain rule. The chain rule states that if we have a function $f(u)$ where $u$ is a function of $x$, then the derivative of $f(u)$ with respect to $x$ is given by $f'(u) \cdot u'(x)$.

Let's break down the function $F(x)$:

$F(x) = \sqrt{(2x + 3)^4}$

Here, $u = (2x + 3)^4$, and $f(u) = \sqrt{u} = u^{1/2}$.

Now, let's find the derivatives:

1. Find $u'(x)$:

$u = (2x + 3)^4$

To find $u'(x)$, we apply the chain rule. The derivative of $u$ with respect to $x$ is:

$u'(x) = 4(2x + 3)^3 \cdot \frac{d}{dx}(2x + 3)$

$u'(x) = 4(2x + 3)^3 \cdot 2$

$u'(x) = 8(2x + 3)^3$

2. Find $f'(u)$:

$f(u) = \sqrt{u} = u^{1/2}$

To find $f'(u)$, we differentiate $f(u)$ with respect to $u$:

$f'(u) = \frac{d}{du}(u^{1/2})$

$f'(u) = \frac{1}{2}u^{-1/2}$

3. Now, we combine the results from step 1 and step 2:

$F'(x) = f'(u) \cdot u'(x)$

$F'(x) = \frac{1}{2}(2x + 3)^{-1/2} \cdot 8(2x + 3)^3$

Finally, we simplify:

$F'(x) = 4(2x + 3)^{3/2}$