Question

F'(x)-?a) f(x)=(x^2-5)*√xb) f(x)=√x/x-1

Answer 1

вот эти производные )))

Answer 2

To find the derivatives of the given functions, we can use the rules of differentiation. Let's find the derivatives of each function:

a) $f(x) = (x^2 - 5) \sqrt{x}$

We'll use the product rule for differentiation, which states that $(uv)' = u'v + uv'$, where $u$ and $v$ are functions of $x$.

Let $u(x) = x^2 - 5$ and $v(x) = \sqrt{x}$.

Now, we need to find the derivatives of $u(x)$ and $v(x)$:

$u'(x) = 2x$ (derivative of $x^2 - 5$) $v'(x) = \frac{1}{2\sqrt{x}}$ (derivative of $\sqrt{x}$)

Now, apply the product rule:

$f'(x) = u'v + uv' = (2x)(\sqrt{x}) + (x^2 - 5)\left(\frac{1}{2\sqrt{x}}\right)$

Simplify:

$f'(x) = 2x\sqrt{x} + \frac{x^2 - 5}{2\sqrt{x}}$

So, the derivative of $f(x)$ with respect to $x$ is:

$f'(x) = 2x\sqrt{x} + \frac{x^2 - 5}{2\sqrt{x}}$

b) $f(x) = \frac{\sqrt{x}}{x - 1}$

To find the derivative of this function, we can use the quotient rule, which states that $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$, where $u$ and $v$ are functions of $x$.

Let $u(x) = \sqrt{x}$ and $v(x) = x - 1$.

Now, we need to find the derivatives of $u(x)$ and (v(x):

$u'(x) = \frac{1}{2\sqrt{x}}$ (derivative of $\sqrt{x}$) $v'(x) = 1$ (derivative of $x - 1$)

Apply the quotient rule:

$f'(x) = \frac{u'v - uv'}{v^2} = \frac{\frac{1}{2\sqrt{x}}(x - 1) - \sqrt{x}(1)}{(x - 1)^2}$