Solve frac{5^{3}x^{4}y^{7}*5^{10}xy^{3}}{5^2xy*5x^3y^7*5^7}

$$\frac { 5 ^ { 3 } x ^ { 4 } y ^ { 7 } \times 5 ^ { 10 } x y ^ { 3 } } { 5 ^ { 2 } x y \times 5 x ^ { 3 } y ^ { 7 } \times 5 ^ { 7 } }$$

Evaluate

$$125xy^{2}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 10 to get 13.

$$\frac{5^{13}x^{4}y^{7}xy^{3}}{5^{2}xy\times 5x^{3}y^{7}\times 5^{7}}$$

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

$$\frac{5^{13}x^{5}y^{7}y^{3}}{5^{2}xy\times 5x^{3}y^{7}\times 5^{7}}$$

To multiply powers of the same base, add their exponents. Add 7 and 3 to get 10.

$$\frac{5^{13}x^{5}y^{10}}{5^{2}xy\times 5x^{3}y^{7}\times 5^{7}}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{5^{13}x^{5}y^{10}}{5^{3}xyx^{3}y^{7}\times 5^{7}}$$

To multiply powers of the same base, add their exponents. Add 3 and 7 to get 10.

$$\frac{5^{13}x^{5}y^{10}}{5^{10}xyx^{3}y^{7}}$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\frac{5^{13}x^{5}y^{10}}{5^{10}x^{4}yy^{7}}$$

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

$$\frac{5^{13}x^{5}y^{10}}{5^{10}x^{4}y^{8}}$$

Cancel out 5^{10}x^{4}y^{8} in both numerator and denominator.

$$5^{3}xy^{2}$$

Calculate 5 to the power of 3 and get 125.

$$125xy^{2}$$

Differentiate w.r.t. x

$$125y^{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1220703125xy^{10}}{9765625x^{3}y^{8}}x^{4-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{125y^{2}}{x^{2}}x^{3})$$

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}.

$$3\times \frac{125y^{2}}{x^{2}}x^{3-1}$$

Do the arithmetic.

$$\frac{375y^{2}}{x^{2}}x^{2}$$