Step-by-step Solution

Find the derivative using the product rule $\frac{d}{dr}\left(\pi \left(3+r\right)\sqrt{\left(3-r\right)^2+\left(\frac{108}{9+r^2+3r}\right)^2}\right)$

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Step-by-step explanation

Problem to solve:

$\frac{d}{dr}\pi\left(3+r\right)\sqrt{\left(3-r\right)^2+\left(\frac{108}{\left(9+r^2+3r\right)}\right)^2}$

Learn how to solve product rule of differentiation problems step by step online.

$\frac{d}{dr}\left(\pi \left(3+r\right)\sqrt{\left(3-r\right)^2+\frac{11664}{\left(9+r^2+3r\right)^2}}\right)$

Unlock this full step-by-step solution!

Learn how to solve product rule of differentiation problems step by step online. Find the derivative using the product rule (d/dr)(pi(3+r)*((3-r)^2+(108/(9+r^2+3r))^2)^0.5). The power of a quotient is equal to the quotient of the power of the numerator and denominator: \displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}. The derivative of a function multiplied by a constant (\pi ) is equal to the constant times the derivative of the function. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=3+r and g=\sqrt{\left(3-r\right)^2+\frac{11664}{\left(9+r^2+3r\right)^2}}. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}.

Final Answer

$\frac{\pi \sqrt{11664+\left(3-r\right)^2\left(9+r^2+3r\right)^2}}{9+r^2+3r}+\frac{\frac{\pi}{2}\left(-23328\left(2r+3\right)-2\left(3-r\right)\left(9+r^2+3r\right)^{3}\right)\left(3+r\right)}{\sqrt{11664+\left(3-r\right)^2\left(9+r^2+3r\right)^2}\left(9+r^2+3r\right)^{2}}$

Problem Analysis

$\frac{d}{dr}\pi\left(3+r\right)\sqrt{\left(3-r\right)^2+\left(\frac{108}{\left(9+r^2+3r\right)}\right)^2}$

Related formulas:

7. See formulas

Time to solve it:

~ 1.03 seconds