Question

Найдите производную функции f(x) = (5+3х )^3. 1) 3(5+3 х )^2 2) (5+3 х )^2 3) -3(5+3 х )^3 4) -(5+3 х )^2.

Answer 1

f(x) = (5+3х )^3.

f'(x) = ((5+3х )^3)'=3(5+3х)^2*(5+3х)'=3(5+3х)^2*3=9(5+3х)^2

Answer 2

To find the derivative of the function $f(x) = (5+3x)^3$ with respect to $x$, we can use the chain rule of differentiation. The chain rule states that if we have a function $g(u)$ and another function $h(x)$ defined as $h(x) = g(u(x))$, then the derivative of $h(x)$ with respect to $x$ is given by $\frac{dh}{dx} = \frac{dg}{du} \cdot \frac{du}{dx}$.

In this case, we can let $u = 5+3x$, and then $f(x) = u^3$. Now, we can find $\frac{df}{dx}$ as follows:

$\frac{df}{dx} = \frac{d}{dx}(u^3) = \frac{d}{du}(u^3) \cdot \frac{du}{dx}$.

The derivative of $u^3$ with respect to $u$ is $3u^2$, and the derivative of $u = 5+3x$ with respect to $x$ is $3$.

Now, multiply the two derivatives together:

$\frac{df}{dx} = 3u^2 \cdot 3 = 9u^2$.

Finally, substitute back $u = 5+3x$ to get the final result:

$\frac{df}{dx} = 9(5+3x)^2$.

So, the correct answer is option 2) $9(5+3x)^2$.