Solve (x^2/(xy^3)(x))

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

Evaluate

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

Solution Steps

Multiply x and x to get x^{2}.

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

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

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

Differentiate w.r.t. y

$$-\frac{3}{y^{4}}$$

Steps Using Chain Rule

Multiply x and x to get x^{2}.

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

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

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

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).

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

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}.

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

Simplify.

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