Solve 12r^-6s^3/3r^3s^-9

$$\frac { 12 r ^ { - 6 } s ^ { 3 } } { 3 r ^ { 3 } s ^ { - 9 } }$$

Evaluate

$$\frac{4s^{12}}{r^{9}}$$

Solution Steps

Cancel out 3 in both numerator and denominator.

$$\frac{4r^{-6}s^{3}}{s^{-9}r^{3}}$$

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

$$\frac{4r^{-6}s^{12}}{r^{3}}$$

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

$$\frac{4s^{12}}{r^{9}}$$

Differentiate w.r.t. r

$$-\frac{36s^{12}}{r^{10}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{12s^{3}s^{9}}{3}r^{-6-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}r}(4s^{12}r^{-9})$$

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}.

$$-9\times 4s^{12}r^{-9-1}$$

Do the arithmetic.

$$\left(-36s^{12}\right)r^{-10}$$