Solve 4x^2y^3/8x^2y^7

$$\frac { 4 x ^ { 2 } y ^ { 3 } } { 8 x ^ { 2 } y ^ { 7 } }$$

Evaluate

$$\frac{1}{2y^{4}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}x^{2}y^{3}}{8^{1}x^{2}y^{7}}$$

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

$$\frac{4^{1}}{8^{1}}x^{2-2}y^{3-7}$$

Subtract 2 from 2.

$$\frac{4^{1}}{8^{1}}x^{0}y^{3-7}$$

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

$$\frac{4^{1}}{8^{1}}y^{3-7}$$

Subtract 7 from 3.

$$\frac{4^{1}}{8^{1}}y^{-4}$$

Reduce the fraction \frac{4}{8} to lowest terms by extracting and canceling out 4.

$$\frac{1}{2}\times \frac{1}{y^{4}}$$

Differentiate w.r.t. y

$$-\frac{2}{y^{5}}$$

Steps Using Chain Rule

Cancel out 4x^{2}y^{3} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{2y^{4}})$$

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).

$$-\left(2y^{4}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}y}(2y^{4})$$

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}.

$$-\left(2y^{4}\right)^{-2}\times 4\times 2y^{4-1}$$

Simplify.

$$-8y^{3}\times \left(2y^{4}\right)^{-2}$$