Solve (x/4pivarepsilon_{0})*(x^2/4pivarepsilon_{0})

$$( \frac { x } { 4 \pi \varepsilon _ { 0 } } ) \times ( \frac { x ^ { 2 } } { 4 \pi \varepsilon _ { 0 } } )$$

Evaluate

$$\frac{x^{3}}{16\left(\pi \epsilon _{0}\right)^{2}}$$

Solution Steps

Multiply \frac{x}{4\pi \epsilon _{0}} times \frac{x^{2}}{4\pi \epsilon _{0}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{xx^{2}}{4\pi \epsilon _{0}\times 4\pi \epsilon _{0}}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{x^{3}}{4\pi \epsilon _{0}\times 4\pi \epsilon _{0}}$$

Multiply \pi and \pi to get \pi ^{2}.

$$\frac{x^{3}}{4\pi ^{2}\epsilon _{0}\times 4\epsilon _{0}}$$

Multiply \epsilon _{0} and \epsilon _{0} to get \epsilon _{0}^{2}.

$$\frac{x^{3}}{4\pi ^{2}\epsilon _{0}^{2}\times 4}$$

Multiply 4 and 4 to get 16.

$$\frac{x^{3}}{16\pi ^{2}\epsilon _{0}^{2}}$$

Differentiate w.r.t. x

$$\frac{3\times \left(\frac{x}{\pi \epsilon _{0}}\right)^{2}}{16}$$