Solve xy^-1*4xy^0/3x^4y^4

$$\frac { x y ^ { - 1 } \cdot 4 x y ^ { 0 } } { 3 x ^ { 4 } y ^ { 4 } }$$

Evaluate

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

Solution Steps

Multiply x and x to get x^{2}.

$$\frac{x^{2}y^{-1}\times 4y^{0}}{3x^{4}y^{4}}$$

To multiply powers of the same base, add their exponents. Add -1 and 0 to get -1.

$$\frac{x^{2}y^{-1}\times 4}{3x^{4}y^{4}}$$

Cancel out x^{2} in both numerator and denominator.

$$\frac{4\times \frac{1}{y}}{3x^{2}y^{4}}$$

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

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4xy^{0}}{y\times 3y^{4}}x^{1-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x}{3y^{5}}x^{-3})$$

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}.

$$-3\times \frac{4x}{3y^{5}}x^{-3-1}$$

Do the arithmetic.

$$\left(-\frac{4x}{y^{5}}\right)x^{-4}$$