Solve frac{12x^{9}y^9}{15x^7y*5x^4y^2}

$$\frac { 12 x ^ { 9 } y ^ { 9 } } { 15 x ^ { 7 } y \cdot 5 x ^ { 4 } y ^ { 2 } }$$

Evaluate

$$\frac{4y^{6}}{25x^{2}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 7 and 4 to get 11.

$$\frac{12x^{9}y^{9}}{15x^{11}y\times 5y^{2}}$$

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

$$\frac{12x^{9}y^{9}}{15x^{11}y^{3}\times 5}$$

Cancel out 3y^{3}x^{9} in both numerator and denominator.

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

Multiply 5 and 5 to get 25.

$$\frac{4y^{6}}{25x^{2}}$$

Differentiate w.r.t. x

$$-\frac{8y^{6}}{25x^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12y^{9}}{75y^{3}x^{4}}x^{9-7})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4y^{6}}{25x^{4}}x^{2})$$

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}.

$$2\times \frac{4y^{6}}{25x^{4}}x^{2-1}$$

Do the arithmetic.

$$\frac{8y^{6}}{25x^{4}}x^{1}$$

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

$$\frac{8y^{6}}{25x^{4}}x$$