Solve 3x^7y/27x^7y^7

$$\frac { 3 x ^ { 7 } y } { 27 x ^ { 7 } y ^ { 7 } }$$

Evaluate

$$\frac{1}{9y^{6}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{3^{1}x^{7}y^{1}}{27^{1}x^{7}y^{7}}$$

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

$$\frac{3^{1}}{27^{1}}x^{7-7}y^{1-7}$$

Subtract 7 from 7.

$$\frac{3^{1}}{27^{1}}x^{0}y^{1-7}$$

For any number a except 0, a^{0}=1.

$$\frac{3^{1}}{27^{1}}y^{1-7}$$

Subtract 7 from 1.

$$\frac{3^{1}}{27^{1}}y^{-6}$$

Reduce the fraction \frac{3}{27} to lowest terms by extracting and canceling out 3.

$$\frac{1}{9}\times \frac{1}{y^{6}}$$

Differentiate w.r.t. y

$$-\frac{2}{3y^{7}}$$

Steps Using Chain Rule

Cancel out 3yx^{7} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{9y^{6}})$$

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).

$$-\left(9y^{6}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}y}(9y^{6})$$

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}.

$$-\left(9y^{6}\right)^{-2}\times 6\times 9y^{6-1}$$

Simplify.

$$-54y^{5}\times \left(9y^{6}\right)^{-2}$$