Solve frac{r^{9}s^{2}t^{0}}{r^{-84}s^{3}t^0*r^12s^4t^5}

$$\frac { r ^ { 9 } s ^ { 2 } t ^ { 0 } } { r ^ { - 84 } s ^ { 3 } t ^ { 0 } \cdot r ^ { 12 } s ^ { 4 } t ^ { 5 } }$$

Evaluate

$$\frac{r^{81}}{\left(st\right)^{5}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -84 and 12 to get -72.

$$\frac{r^{9}s^{2}t^{0}}{r^{-72}s^{3}t^{0}s^{4}t^{5}}$$

To multiply powers of the same base, add their exponents. Add 3 and 4 to get 7.

$$\frac{r^{9}s^{2}t^{0}}{r^{-72}s^{7}t^{0}t^{5}}$$

To multiply powers of the same base, add their exponents. Add 0 and 5 to get 5.

$$\frac{r^{9}s^{2}t^{0}}{r^{-72}s^{7}t^{5}}$$

Cancel out s^{2} in both numerator and denominator.

$$\frac{t^{0}r^{9}}{r^{-72}s^{5}t^{5}}$$

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

$$\frac{t^{0}r^{81}}{s^{5}t^{5}}$$

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

$$\frac{r^{81}}{s^{5}t^{5}}$$

Differentiate w.r.t. r

$$\frac{81r^{80}}{\left(st\right)^{5}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{t^{0}s^{2}}{t^{0}s^{3}s^{4}t^{5}r^{12}}r^{9-\left(-84\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{r^{12}\left(st\right)^{5}}r^{93})$$

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}.

$$93\times \frac{1}{r^{12}\left(st\right)^{5}}r^{93-1}$$

Do the arithmetic.

$$\frac{93}{r^{12}\left(st\right)^{5}}r^{92}$$