Question

U(x)=x^(1/2), v(x)=3x-2, f(x)=u(v(x)). решить уравнение f'(x)=0,375

Answer 1

U(x)=x¹/²
v(x)=3x-2

f(x)=u(v(x))=(v(x))¹/²=(3x-2)¹/²

f'(x)=1/2*(3x-2)^(-1/2)*(3x-2)'=
3/2*1/(3x-2)¹/²

3/2*1/(3x-2)¹/²=375/1000=15/40=3/20
1/(3x-2)¹/²=3/20*2/3=1/10
(3x-2)¹/²=10
3x-2=100
3x=102
x=102:3=34

Answer 2

To find the solution to the equation $f'(x) = 0.375$, we need to first calculate the derivative of $f(x)$. Let's start by expressing $f(x)$ as a composition of functions $u(x)$ and $v(x)$:

$f(x) = u(v(x)) = \sqrt{3x-2}$

Now, let's find the derivative of $f(x)$ with respect to $x$:

$f'(x) = \frac{d}{dx} \sqrt{3x-2}$

Using the chain rule, the derivative of $\sqrt{3x-2}$ is:

$f'(x) = \frac{1}{2\sqrt{3x-2}} \cdot \frac{d}{dx} (3x-2) = \frac{1}{2\sqrt{3x-2}} \cdot 3 = \frac{3}{2\sqrt{3x-2}}$

Now, we need to solve the equation $f'(x) = 0.375$:

$\frac{3}{2\sqrt{3x-2}} = 0.375$

To solve for $x$, we can isolate $\sqrt{3x-2}$ and then square both sides to eliminate the square root:

$2\sqrt{3x-2} = \frac{3}{0.375} = 8$

Now, divide both sides by 2:

$\sqrt{3x-2} = 4$

Finally, square both sides to find the value of $x$:

$3x-2 = 4^2 = 16$

Now, add 2 to both sides:

$3x = 16 + 2 = 18$

And, divide both sides by 3:

$x = \frac{18}{3} = 6$

So, the solution to the equation $f'(x) = 0.375$ is $x = 6$.