Solve (frac{2*10^{-6}c*-2*10^{-6}c}{0.3}+-2*10^-6c*1*10^-6c/0.3+2*10^-6c*1*10^-6c/0.3)

$$( \frac { 2 \times 10 ^ { - 6 } c \cdot - 2 \times 10 ^ { - 6 } c } { 0.3 } + \frac { - 2 \times 10 ^ { - 6 } c \cdot 1 \times 10 ^ { - 6 } c } { 0.3 } + \frac { 2 \times 10 ^ { - 6 } c \cdot 1 \times 10 ^ { - 6 } c } { 0.3 } )$$

Evaluate

$$-\frac{c^{2}}{75000000000}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -6 and -6 to get -12.

$$\frac{2\times 10^{-12}c\left(-2\right)c}{0.3}+\frac{-2\times 10^{-6}c\times 10^{-6}c}{0.3}+\frac{2\times 10^{-6}c\times 1\times 10^{-6}c}{0.3}$$

Multiply c and c to get c^{2}.

$$\frac{2\times 10^{-12}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-6}c\times 10^{-6}c}{0.3}+\frac{2\times 10^{-6}c\times 1\times 10^{-6}c}{0.3}$$

To multiply powers of the same base, add their exponents. Add -6 and -6 to get -12.

$$\frac{2\times 10^{-12}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-12}cc}{0.3}+\frac{2\times 10^{-6}c\times 1\times 10^{-6}c}{0.3}$$

Multiply c and c to get c^{2}.

$$\frac{2\times 10^{-12}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-6}c\times 1\times 10^{-6}c}{0.3}$$

To multiply powers of the same base, add their exponents. Add -6 and -6 to get -12.

$$\frac{2\times 10^{-12}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-12}c\times 1c}{0.3}$$

Multiply c and c to get c^{2}.

$$\frac{2\times 10^{-12}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3}$$

Calculate 10 to the power of -12 and get \frac{1}{1000000000000}.

$$\frac{2\times \frac{1}{1000000000000}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3}$$

Multiply 2 and \frac{1}{1000000000000} to get \frac{1}{500000000000}.

$$\frac{\frac{1}{500000000000}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3}$$

Multiply \frac{1}{500000000000} and -2 to get -\frac{1}{250000000000}.

$$\frac{-\frac{1}{250000000000}c^{2}}{0.3}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3}$$

Divide -\frac{1}{250000000000}c^{2} by 0.3 to get -\frac{1}{75000000000}c^{2}.

$$-\frac{1}{75000000000}c^{2}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3}$$

Calculate 10 to the power of -12 and get \frac{1}{1000000000000}.

$$-\frac{1}{75000000000}c^{2}+\frac{-2\times \frac{1}{1000000000000}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3}$$

Multiply -2 and \frac{1}{1000000000000} to get -\frac{1}{500000000000}.

$$-\frac{1}{75000000000}c^{2}+\frac{-\frac{1}{500000000000}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3}$$

Divide -\frac{1}{500000000000}c^{2} by 0.3 to get -\frac{1}{150000000000}c^{2}.

$$-\frac{1}{75000000000}c^{2}-\frac{1}{150000000000}c^{2}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3}$$

Combine -\frac{1}{75000000000}c^{2} and -\frac{1}{150000000000}c^{2} to get -\frac{1}{50000000000}c^{2}.

$$-\frac{1}{50000000000}c^{2}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3}$$

Calculate 10 to the power of -12 and get \frac{1}{1000000000000}.

$$-\frac{1}{50000000000}c^{2}+\frac{2\times \frac{1}{1000000000000}c^{2}\times 1}{0.3}$$

Multiply 2 and \frac{1}{1000000000000} to get \frac{1}{500000000000}.

$$-\frac{1}{50000000000}c^{2}+\frac{\frac{1}{500000000000}c^{2}\times 1}{0.3}$$

Multiply \frac{1}{500000000000} and 1 to get \frac{1}{500000000000}.

$$-\frac{1}{50000000000}c^{2}+\frac{\frac{1}{500000000000}c^{2}}{0.3}$$

Divide \frac{1}{500000000000}c^{2} by 0.3 to get \frac{1}{150000000000}c^{2}.

$$-\frac{1}{50000000000}c^{2}+\frac{1}{150000000000}c^{2}$$

Combine -\frac{1}{50000000000}c^{2} and \frac{1}{150000000000}c^{2} to get -\frac{1}{75000000000}c^{2}.

$$-\frac{1}{75000000000}c^{2}$$

Differentiate w.r.t. c

$$-\frac{c}{37500000000}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add -6 and -6 to get -12.

$$\frac{\mathrm{d}}{\mathrm{d}c}(\frac{2\times 10^{-12}c\left(-2\right)c}{0.3}+\frac{-2\times 10^{-6}c\times 10^{-6}c}{0.3}+\frac{2\times 10^{-6}c\times 1\times 10^{-6}c}{0.3})$$

Multiply c and c to get c^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(\frac{2\times 10^{-12}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-6}c\times 10^{-6}c}{0.3}+\frac{2\times 10^{-6}c\times 1\times 10^{-6}c}{0.3})$$

To multiply powers of the same base, add their exponents. Add -6 and -6 to get -12.

$$\frac{\mathrm{d}}{\mathrm{d}c}(\frac{2\times 10^{-12}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-12}cc}{0.3}+\frac{2\times 10^{-6}c\times 1\times 10^{-6}c}{0.3})$$

Multiply c and c to get c^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(\frac{2\times 10^{-12}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-6}c\times 1\times 10^{-6}c}{0.3})$$

To multiply powers of the same base, add their exponents. Add -6 and -6 to get -12.

$$\frac{\mathrm{d}}{\mathrm{d}c}(\frac{2\times 10^{-12}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-12}c\times 1c}{0.3})$$

Multiply c and c to get c^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(\frac{2\times 10^{-12}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3})$$

Calculate 10 to the power of -12 and get \frac{1}{1000000000000}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(\frac{2\times \frac{1}{1000000000000}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3})$$

Multiply 2 and \frac{1}{1000000000000} to get \frac{1}{500000000000}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(\frac{\frac{1}{500000000000}c^{2}\left(-2\right)}{0.3}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3})$$

Multiply \frac{1}{500000000000} and -2 to get -\frac{1}{250000000000}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(\frac{-\frac{1}{250000000000}c^{2}}{0.3}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3})$$

Divide -\frac{1}{250000000000}c^{2} by 0.3 to get -\frac{1}{75000000000}c^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(-\frac{1}{75000000000}c^{2}+\frac{-2\times 10^{-12}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3})$$

Calculate 10 to the power of -12 and get \frac{1}{1000000000000}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(-\frac{1}{75000000000}c^{2}+\frac{-2\times \frac{1}{1000000000000}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3})$$

Multiply -2 and \frac{1}{1000000000000} to get -\frac{1}{500000000000}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(-\frac{1}{75000000000}c^{2}+\frac{-\frac{1}{500000000000}c^{2}}{0.3}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3})$$

Divide -\frac{1}{500000000000}c^{2} by 0.3 to get -\frac{1}{150000000000}c^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(-\frac{1}{75000000000}c^{2}-\frac{1}{150000000000}c^{2}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3})$$

Combine -\frac{1}{75000000000}c^{2} and -\frac{1}{150000000000}c^{2} to get -\frac{1}{50000000000}c^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(-\frac{1}{50000000000}c^{2}+\frac{2\times 10^{-12}c^{2}\times 1}{0.3})$$

Calculate 10 to the power of -12 and get \frac{1}{1000000000000}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(-\frac{1}{50000000000}c^{2}+\frac{2\times \frac{1}{1000000000000}c^{2}\times 1}{0.3})$$

Multiply 2 and \frac{1}{1000000000000} to get \frac{1}{500000000000}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(-\frac{1}{50000000000}c^{2}+\frac{\frac{1}{500000000000}c^{2}\times 1}{0.3})$$

Multiply \frac{1}{500000000000} and 1 to get \frac{1}{500000000000}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(-\frac{1}{50000000000}c^{2}+\frac{\frac{1}{500000000000}c^{2}}{0.3})$$

Divide \frac{1}{500000000000}c^{2} by 0.3 to get \frac{1}{150000000000}c^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(-\frac{1}{50000000000}c^{2}+\frac{1}{150000000000}c^{2})$$

Combine -\frac{1}{50000000000}c^{2} and \frac{1}{150000000000}c^{2} to get -\frac{1}{75000000000}c^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(-\frac{1}{75000000000}c^{2})$$

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

$$2\left(-\frac{1}{75000000000}\right)c^{2-1}$$

Multiply 2 times -\frac{1}{75000000000}.

$$-\frac{1}{37500000000}c^{2-1}$$

Subtract 1 from 2.

$$-\frac{1}{37500000000}c^{1}$$

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

$$-\frac{1}{37500000000}c$$