Solve x/2y2/3xy^2

$$\frac { x } { 2 y } \frac { 2 } { 3 x y ^ { 2 } }$$

Evaluate

$$\frac{1}{3y^{3}}$$

Solution Steps

Multiply \frac{x}{2y} times \frac{2}{3xy^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x\times 2}{2y\times 3xy^{2}}$$

Cancel out 2x in both numerator and denominator.

$$\frac{1}{3yy^{2}}$$

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

$$\frac{1}{3y^{3}}$$

Differentiate w.r.t. y

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

Steps Using Chain Rule

Multiply \frac{x}{2y} times \frac{2}{3xy^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{x\times 2}{2y\times 3xy^{2}})$$

Cancel out 2x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{3yy^{2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{3y^{3}})$$

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).

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

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}.

$$-\left(3y^{3}\right)^{-2}\times 3\times 3y^{3-1}$$

Simplify.

$$-9y^{2}\times \left(3y^{3}\right)^{-2}$$