Solve frac{36x^{7}y^{-3}(x^-2y^5)}{9x^5y^-4}

$$\frac { 36 x ^ { 7 } y ^ { - 3 } ( x ^ { - 2 } y ^ { 5 } ) } { 9 x ^ { 5 } y ^ { - 4 } }$$

Evaluate

$$4y^{6}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 7 and -2 to get 5.

$$\frac{36x^{5}y^{-3}y^{5}}{9x^{5}y^{-4}}$$

To multiply powers of the same base, add their exponents. Add -3 and 5 to get 2.

$$\frac{36x^{5}y^{2}}{9x^{5}y^{-4}}$$

Cancel out 9x^{5} in both numerator and denominator.

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

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

$$4y^{6}$$

Differentiate w.r.t. y

$$24y^{5}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 7 and -2 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{36x^{5}y^{-3}y^{5}}{9x^{5}y^{-4}})$$

To multiply powers of the same base, add their exponents. Add -3 and 5 to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{36x^{5}y^{2}}{9x^{5}y^{-4}})$$

Cancel out 9x^{5} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{4y^{2}}{y^{-4}})$$

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

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

The derivative of ax^{n} is nax^{n-1}.

$$6\times 4y^{6-1}$$

Multiply 6 times 4.

$$24y^{6-1}$$

Subtract 1 from 6.

$$24y^{5}$$