Solve -8c^-5d^-4/-48cd^-12

$$\frac { - 8 c ^ { - 5 } d ^ { - 4 } } { - 48 c d ^ { - 12 } }$$

Evaluate

$$\frac{d^{8}}{6c^{6}}$$

Solution Steps

Cancel out 8 in both numerator and denominator.

$$\frac{-c^{-5}d^{-4}}{-6d^{-12}c}$$

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

$$\frac{-c^{-5}d^{8}}{-6c}$$

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

$$\frac{-d^{8}}{-6c^{6}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{d^{8}}{6c^{6}}$$

Differentiate w.r.t. c

$$-\frac{d^{8}}{c^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}c}(\left(-\frac{\frac{8}{d^{4}}}{-\frac{48}{d^{12}}}\right)c^{-5-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}c}(\frac{d^{8}}{6}c^{-6})$$

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}.

$$-6\times \frac{d^{8}}{6}c^{-6-1}$$

Do the arithmetic.

$$\left(-d^{8}\right)c^{-7}$$