Solve frac{(x^{6})^{2}(x^{3}y)^{-2}(xy^0)}{(x^2y^2)^3xy}

$$\frac { ( x ^ { 6 } ) ^ { 2 } ( x ^ { 3 } y ) ^ { - 2 } ( x y ^ { 0 } ) } { ( x ^ { 2 } y ^ { 2 } ) ^ { 3 } x y }$$

Evaluate

$$\frac{1}{y^{9}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 6 and 2 to get 12.

$$\frac{x^{12}\left(x^{3}y\right)^{-2}xy^{0}}{\left(x^{2}y^{2}\right)^{3}xy}$$

To multiply powers of the same base, add their exponents. Add 12 and 1 to get 13.

$$\frac{x^{13}\left(x^{3}y\right)^{-2}y^{0}}{\left(x^{2}y^{2}\right)^{3}xy}$$

Cancel out x in both numerator and denominator.

$$\frac{\left(yx^{3}\right)^{-2}y^{0}x^{12}}{y\left(x^{2}y^{2}\right)^{3}}$$

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

$$\frac{\left(yx^{3}\right)^{-2}x^{12}}{y^{1}\left(x^{2}y^{2}\right)^{3}}$$

Expand \left(yx^{3}\right)^{-2}.

$$\frac{y^{-2}\left(x^{3}\right)^{-2}x^{12}}{y^{1}\left(x^{2}y^{2}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -2 to get -6.

$$\frac{y^{-2}x^{-6}x^{12}}{y^{1}\left(x^{2}y^{2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add -6 and 12 to get 6.

$$\frac{y^{-2}x^{6}}{y^{1}\left(x^{2}y^{2}\right)^{3}}$$

Calculate y to the power of 1 and get y.

$$\frac{y^{-2}x^{6}}{y\left(x^{2}y^{2}\right)^{3}}$$

Expand \left(x^{2}y^{2}\right)^{3}.

$$\frac{y^{-2}x^{6}}{y\left(x^{2}\right)^{3}\left(y^{2}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{y^{-2}x^{6}}{yx^{6}\left(y^{2}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{y^{-2}x^{6}}{yx^{6}y^{6}}$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\frac{y^{-2}x^{6}}{y^{7}x^{6}}$$

Cancel out x^{6} in both numerator and denominator.

$$\frac{y^{-2}}{y^{7}}$$

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

$$\frac{1}{y^{9}}$$

Differentiate w.r.t. y

$$-\frac{9}{y^{10}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 6 and 2 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{x^{12}\left(x^{3}y\right)^{-2}xy^{0}}{\left(x^{2}y^{2}\right)^{3}xy})$$

To multiply powers of the same base, add their exponents. Add 12 and 1 to get 13.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{x^{13}\left(x^{3}y\right)^{-2}y^{0}}{\left(x^{2}y^{2}\right)^{3}xy})$$

Cancel out x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\left(yx^{3}\right)^{-2}y^{0}x^{12}}{y\left(x^{2}y^{2}\right)^{3}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\left(yx^{3}\right)^{-2}x^{12}}{y^{1}\left(x^{2}y^{2}\right)^{3}})$$

Expand \left(yx^{3}\right)^{-2}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{-2}\left(x^{3}\right)^{-2}x^{12}}{y^{1}\left(x^{2}y^{2}\right)^{3}})$$

To raise a power to another power, multiply the exponents. Multiply 3 and -2 to get -6.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{-2}x^{-6}x^{12}}{y^{1}\left(x^{2}y^{2}\right)^{3}})$$

To multiply powers of the same base, add their exponents. Add -6 and 12 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{-2}x^{6}}{y^{1}\left(x^{2}y^{2}\right)^{3}})$$

Calculate y to the power of 1 and get y.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{-2}x^{6}}{y\left(x^{2}y^{2}\right)^{3}})$$

Expand \left(x^{2}y^{2}\right)^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{-2}x^{6}}{y\left(x^{2}\right)^{3}\left(y^{2}\right)^{3}})$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{-2}x^{6}}{yx^{6}\left(y^{2}\right)^{3}})$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{-2}x^{6}}{yx^{6}y^{6}})$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{-2}x^{6}}{y^{7}x^{6}})$$

Cancel out x^{6} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{-2}}{y^{7}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{y^{9}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(y^{9}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}y}(y^{9})$$

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}.

$$-\left(y^{9}\right)^{-2}\times 9y^{9-1}$$

Simplify.

$$-9y^{8}\left(y^{9}\right)^{-2}$$