Find the derivative of the function below by simplifying ?
F(x) =( x - 7x √x) ÷ √x

Question

Cameron Alexander · Accepted Answer

#f'(x)=(1-14sqrtx)/(2sqrtx)#

#f(x)=(x-7xsqrtx)/sqrtx#

The fraction can be divided into two separate terms:

#x/sqrtx=(sqrtxsqrtx)/sqrtx=sqrtx#

#(7xsqrtx)/sqrtx=7x#

So:

#f(x)=sqrtx-7x#

which the power rule can be used to distinguish:

#f(x)=x^(1/2)-7x^1#

#f'(x)=1/2x^(-1/2)-7(1)x^0#

#f'(x)=1/(2sqrtx)-7#

identifying a shared denominator

#f'(x)=(1-14sqrtx)/(2sqrtx)#

Ezekiel Alexander · Answer

undefined To find the derivative of the given function $ F(x) = \frac{x - 7x\sqrt{x}}{\sqrt{x}} $, we use the quotient rule. The quotient rule states that if $ u(x) $ and $ v(x) $ are differentiable functions of $ x $, then the derivative of $ \frac{u(x)}{v(x)} $ with respect to $ x $ is given by:

$$ \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} $$

Let $ u(x) = x - 7x\sqrt{x} $ and $ v(x) = \sqrt{x} $. Then, we find the derivatives:

$$ u'(x) = 1 - \frac{7}{2\sqrt{x}} $$
$$ v'(x) = \frac{1}{2\sqrt{x}} $$

Now, applying the quotient rule:

$$ F'(x) = \frac{(1 - \frac{7}{2\sqrt{x}}) \cdot \sqrt{x} - (x - 7x\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}}{(\sqrt{x})^2} $$

$$ F'(x) = \frac{\sqrt{x} - \frac{7}{2} - \frac{x}{2\sqrt{x}} + \frac{7x}{2}}{x} $$

$$ F'(x) = \frac{\sqrt{x} - \frac{x}{2\sqrt{x}}}{x} $$

$$ F'(x) = \frac{\frac{2x - x}{2\sqrt{x}}}{x} $$

$$ F'(x) = \frac{x}{2x\sqrt{x}} $$

$$ F'(x) = \frac{1}{2\sqrt{x}} $$

So, the derivative of $ F(x) $ with respect to $ x $ is $ \frac{1}{2\sqrt{x}} $.

Find the derivative of the function below by simplifying ? F(x) =( x - 7x √x) ÷ √x