Solve R^21/2*3*14sqrt{5*20*10^-6}

$$R ^ { 2 } \frac { 1 } { 2 \times 3 \cdot 14 \sqrt { 5 \times 20 \times 10 ^ { - 6 } } }$$

Evaluate

$$\frac{\sqrt{10000}R^{2}}{84}$$

Solution Steps

Multiply 2 and 3 to get 6.

$$R^{2}\times \frac{1}{6\times 14\sqrt{5\times 20\times 10^{-6}}}$$

Multiply 6 and 14 to get 84.

$$R^{2}\times \frac{1}{84\sqrt{5\times 20\times 10^{-6}}}$$

Multiply 5 and 20 to get 100.

$$R^{2}\times \frac{1}{84\sqrt{100\times 10^{-6}}}$$

Calculate 10 to the power of -6 and get \frac{1}{1000000}.

$$R^{2}\times \frac{1}{84\sqrt{100\times \frac{1}{1000000}}}$$

Multiply 100 and \frac{1}{1000000} to get \frac{1}{10000}.

$$R^{2}\times \frac{1}{84\sqrt{\frac{1}{10000}}}$$

Rewrite the square root of the division \frac{1}{10000} as the division of square roots \frac{\sqrt{1}}{\sqrt{10000}}. Take the square root of both numerator and denominator.

$$R^{2}\times \frac{1}{84\times \frac{1}{100}}$$

Multiply 84 and \frac{1}{100} to get \frac{21}{25}.

$$R^{2}\times \frac{1}{\frac{21}{25}}$$

Divide 1 by \frac{21}{25} by multiplying 1 by the reciprocal of \frac{21}{25}.

$$R^{2}\times 1\times \frac{25}{21}$$

Multiply 1 and \frac{25}{21} to get \frac{25}{21}.

$$R^{2}\times \frac{25}{21}$$

Differentiate w.r.t. R

$$\frac{50R}{21}$$

Steps Using Definition of a Derivative

Multiply 2 and 3 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}R}(R^{2}\times \frac{1}{6\times 14\sqrt{5\times 20\times 10^{-6}}})$$

Multiply 6 and 14 to get 84.

$$\frac{\mathrm{d}}{\mathrm{d}R}(R^{2}\times \frac{1}{84\sqrt{5\times 20\times 10^{-6}}})$$

Multiply 5 and 20 to get 100.

$$\frac{\mathrm{d}}{\mathrm{d}R}(R^{2}\times \frac{1}{84\sqrt{100\times 10^{-6}}})$$

Calculate 10 to the power of -6 and get \frac{1}{1000000}.

$$\frac{\mathrm{d}}{\mathrm{d}R}(R^{2}\times \frac{1}{84\sqrt{100\times \frac{1}{1000000}}})$$

Multiply 100 and \frac{1}{1000000} to get \frac{1}{10000}.

$$\frac{\mathrm{d}}{\mathrm{d}R}(R^{2}\times \frac{1}{84\sqrt{\frac{1}{10000}}})$$

Rewrite the square root of the division \frac{1}{10000} as the division of square roots \frac{\sqrt{1}}{\sqrt{10000}}. Take the square root of both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}R}(R^{2}\times \frac{1}{84\times \frac{1}{100}})$$

Multiply 84 and \frac{1}{100} to get \frac{21}{25}.

$$\frac{\mathrm{d}}{\mathrm{d}R}(R^{2}\times \frac{1}{\frac{21}{25}})$$

Divide 1 by \frac{21}{25} by multiplying 1 by the reciprocal of \frac{21}{25}.

$$\frac{\mathrm{d}}{\mathrm{d}R}(R^{2}\times 1\times \frac{25}{21})$$

Multiply 1 and \frac{25}{21} to get \frac{25}{21}.

$$\frac{\mathrm{d}}{\mathrm{d}R}(R^{2}\times \frac{25}{21})$$

The derivative of ax^{n} is nax^{n-1}.

$$2\times \frac{25}{21}R^{2-1}$$

Multiply 2 times \frac{25}{21}.

$$\frac{50}{21}R^{2-1}$$

Subtract 1 from 2.

$$\frac{50}{21}R^{1}$$

For any term t, t^{1}=t.

$$\frac{50}{21}R$$