Solve 5/4x^3y^8/-5xy^-2

$$\frac { \frac { 5 } { 4 } x ^ { 3 } y ^ { 8 } } { - 5 x y ^ { - 2 } }$$

Evaluate

$$-\frac{x^{2}y^{10}}{4}$$

Solution Steps

Cancel out x in both numerator and denominator.

$$\frac{\frac{5}{4}x^{2}y^{8}}{-5y^{-2}}$$

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

$$\frac{\frac{5}{4}x^{2}y^{10}}{-5}$$

Divide \frac{5}{4}x^{2}y^{10} by -5 to get -\frac{1}{4}x^{2}y^{10}.

$$-\frac{1}{4}x^{2}y^{10}$$

Differentiate w.r.t. x

$$-\frac{xy^{10}}{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5y^{8}}{4\left(-\frac{5}{y^{2}}\right)}x^{3-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{y^{10}}{4}\right)x^{2})$$

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}.

$$2\left(-\frac{y^{10}}{4}\right)x^{2-1}$$

Do the arithmetic.

$$\left(-\frac{y^{10}}{2}\right)x^{1}$$

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

$$\left(-\frac{y^{10}}{2}\right)x$$