Solve n^-7/n^0*n^-85*n^17

$$\frac { n ^ { - 7 } } { n ^ { 0 } \cdot n ^ { - 85 } \cdot n ^ { 17 } }$$

Evaluate

$$n^{61}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 0 and -85 to get -85.

$$\frac{n^{-7}}{n^{-85}n^{17}}$$

To multiply powers of the same base, add their exponents. Add -85 and 17 to get -68.

$$\frac{n^{-7}}{n^{-68}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -68 from -7 to get 61.

$$n^{61}$$

Differentiate w.r.t. n

$$61n^{60}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{n^{0}n^{17}}n^{-7-\left(-85\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{n^{17}}n^{78})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$78\times \frac{1}{n^{17}}n^{78-1}$$

Do the arithmetic.

$$\frac{78}{n^{17}}n^{77}$$