Solve frac{qr^{2}s^{8}t^{3}}{q^{5}r^{0}s^{6}t^-3*qr^9s^6t^0}

$$\frac { q r ^ { 2 } s ^ { 8 } t ^ { 3 } } { q ^ { 5 } r ^ { 0 } s ^ { 6 } t ^ { - 3 } \cdot q r ^ { 9 } s ^ { 6 } t ^ { 0 } }$$

Evaluate

$$\frac{t^{6}}{s^{4}q^{5}r^{7}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and 1 to get 6.

$$\frac{qr^{2}s^{8}t^{3}}{q^{6}r^{0}s^{6}t^{-3}r^{9}s^{6}t^{0}}$$

To multiply powers of the same base, add their exponents. Add 0 and 9 to get 9.

$$\frac{qr^{2}s^{8}t^{3}}{q^{6}r^{9}s^{6}t^{-3}s^{6}t^{0}}$$

To multiply powers of the same base, add their exponents. Add 6 and 6 to get 12.

$$\frac{qr^{2}s^{8}t^{3}}{q^{6}r^{9}s^{12}t^{-3}t^{0}}$$

To multiply powers of the same base, add their exponents. Add -3 and 0 to get -3.

$$\frac{qr^{2}s^{8}t^{3}}{q^{6}r^{9}s^{12}t^{-3}}$$

Cancel out qr^{2}s^{8} in both numerator and denominator.

$$\frac{t^{3}}{t^{-3}s^{4}q^{5}r^{7}}$$

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

$$\frac{t^{6}}{s^{4}q^{5}r^{7}}$$

Differentiate w.r.t. q

$$-\frac{5\times \left(\frac{t}{q}\right)^{6}}{s^{4}r^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}q}(\frac{r^{2}t^{3}s^{8}t^{3}}{qr^{0}t^{0}s^{6}s^{6}r^{9}}q^{1-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}q}(\frac{t^{6}}{qs^{4}r^{7}}q^{-4})$$

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}.

$$-4\times \frac{t^{6}}{qs^{4}r^{7}}q^{-4-1}$$

Do the arithmetic.

$$\left(-\frac{4t^{6}}{qs^{4}r^{7}}\right)q^{-5}$$