Solve (5xy)(4x^5y^7)/10x^2y^4

$$\frac { ( 5 x y ) ( 4 x ^ { 5 } y ^ { 7 } ) } { 10 x ^ { 2 } y ^ { 4 } }$$

Evaluate

$$2\left(xy\right)^{4}$$

Solution Steps

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

$$\frac{5x^{6}y\times 4y^{7}}{10x^{2}y^{4}}$$

To multiply powers of the same base, add their exponents. Add 1 and 7 to get 8.

$$\frac{5x^{6}y^{8}\times 4}{10x^{2}y^{4}}$$

Cancel out 2\times 5x^{2}y^{4} in both numerator and denominator.

$$2x^{4}y^{4}$$

Differentiate w.r.t. x

$$8x^{3}y^{4}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{20x^{5}y^{8}}{10y^{4}}x^{1-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2y^{4}x^{5}\times \frac{1}{x})$$

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}.

$$-2y^{4}x^{5}x^{-1-1}$$

Do the arithmetic.

$$\left(-2y^{4}x^{5}\right)x^{-2}$$