Solve frac{-3p^{4}q^{-4}r^2}{-3q*p^3q^4r^3}

$$\frac { - 3 p ^ { 4 } q ^ { - 4 } r ^ { 2 } } { - 3 q \cdot p ^ { 3 } q ^ { 4 } r ^ { 3 } }$$

Evaluate

$$\frac{p}{rq^{9}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$\frac{-3p^{4}q^{-4}r^{2}}{-3q^{5}p^{3}r^{3}}$$

Cancel out -3r^{2}p^{3} in both numerator and denominator.

$$\frac{q^{-4}p}{rq^{5}}$$

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

$$\frac{p}{rq^{9}}$$

Differentiate w.r.t. p

$$\frac{1}{rq^{9}}$$

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{1}{rq^{9}}p^{1})$$

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}.

$$\frac{1}{rq^{9}}p^{1-1}$$

Do the arithmetic.

$$\frac{1}{rq^{9}}p^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{1}{rq^{9}}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{1}{rq^{9}}$$