Solve {r}{314*10^-3*1n^frac{278{301}}}{*218-301/278*301}

$$\left. \begin{array} { r } { 314 \times 10 ^ { - 3 } \times 1 n ^ { \frac { 278 } { 301 } } } \\ { \times \frac { 218 - 301 } { 278 \times 301 } } \end{array} \right.$$

Evaluate

$$-\frac{13031n^{\frac{278}{301}}}{41839000}$$

Solution Steps

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$314\times \frac{1}{1000}\times 1n^{\frac{278}{301}}\times \frac{218-301}{278\times 301}$$

Multiply 314 and \frac{1}{1000} to get \frac{157}{500}.

$$\frac{157}{500}\times 1n^{\frac{278}{301}}\times \frac{218-301}{278\times 301}$$

Multiply \frac{157}{500} and 1 to get \frac{157}{500}.

$$\frac{157}{500}n^{\frac{278}{301}}\times \frac{218-301}{278\times 301}$$

Subtract 301 from 218 to get -83.

$$\frac{157}{500}n^{\frac{278}{301}}\times \frac{-83}{278\times 301}$$

Multiply 278 and 301 to get 83678.

$$\frac{157}{500}n^{\frac{278}{301}}\times \frac{-83}{83678}$$

Fraction \frac{-83}{83678} can be rewritten as -\frac{83}{83678} by extracting the negative sign.

$$\frac{157}{500}n^{\frac{278}{301}}\left(-\frac{83}{83678}\right)$$

Multiply \frac{157}{500} and -\frac{83}{83678} to get -\frac{13031}{41839000}.

$$-\frac{13031}{41839000}n^{\frac{278}{301}}$$

Differentiate w.r.t. n

$$-\frac{13031}{45300500n^{\frac{23}{301}}}$$

Steps Using Definition of a Derivative

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(314\times \frac{1}{1000}\times 1n^{\frac{278}{301}}\times \frac{218-301}{278\times 301})$$

Multiply 314 and \frac{1}{1000} to get \frac{157}{500}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{157}{500}\times 1n^{\frac{278}{301}}\times \frac{218-301}{278\times 301})$$

Multiply \frac{157}{500} and 1 to get \frac{157}{500}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{157}{500}n^{\frac{278}{301}}\times \frac{218-301}{278\times 301})$$

Subtract 301 from 218 to get -83.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{157}{500}n^{\frac{278}{301}}\times \frac{-83}{278\times 301})$$

Multiply 278 and 301 to get 83678.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{157}{500}n^{\frac{278}{301}}\times \frac{-83}{83678})$$

Fraction \frac{-83}{83678} can be rewritten as -\frac{83}{83678} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{157}{500}n^{\frac{278}{301}}\left(-\frac{83}{83678}\right))$$

Multiply \frac{157}{500} and -\frac{83}{83678} to get -\frac{13031}{41839000}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(-\frac{13031}{41839000}n^{\frac{278}{301}})$$

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

$$\frac{278}{301}\left(-\frac{13031}{41839000}\right)n^{\frac{278}{301}-1}$$

Multiply \frac{278}{301} times -\frac{13031}{41839000} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-\frac{13031}{45300500}n^{\frac{278}{301}-1}$$

Subtract 1 from \frac{278}{301}.

$$-\frac{13031}{45300500}n^{-\frac{23}{301}}$$