Solve frac{10p^{4}q^{3}*2p^2q^3}{5p^8q^3}

$$\frac { 10 p ^ { 4 } q ^ { 3 } \times 2 p ^ { 2 } q ^ { 3 } } { 5 p ^ { 8 } q ^ { 3 } }$$

Evaluate

$$\frac{4q^{3}}{p^{2}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{10p^{6}q^{3}\times 2q^{3}}{5p^{8}q^{3}}$$

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$\frac{10p^{6}q^{6}\times 2}{5p^{8}q^{3}}$$

Cancel out 5q^{3}p^{6} in both numerator and denominator.

$$\frac{2\times 2q^{3}}{p^{2}}$$

Multiply 2 and 2 to get 4.

$$\frac{4q^{3}}{p^{2}}$$

Differentiate w.r.t. p

$$-8\times \left(\frac{q}{p}\right)^{3}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{20p^{2}q^{6}}{5q^{3}}p^{4-8})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}p}(4p^{2}q^{3}p^{-4})$$

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}.

$$-4\times 4p^{2}q^{3}p^{-4-1}$$

Do the arithmetic.

$$\left(-16p^{2}q^{3}\right)p^{-5}$$