Solve frac{x^{2}+xy-xz}{(x+y)^{2}-z^{2}}divx/(x+z)^{2-y^2}*xy-y^2-yz/(x-y)^2-z^2

$$\frac { x ^ { 2 } + x y - x z } { ( x + y ) ^ { 2 } - z ^ { 2 } } \div \frac { x } { ( x + z ) ^ { 2 } - y ^ { 2 } } \cdot \frac { x y - y ^ { 2 } - y z } { ( x - y ) ^ { 2 } - z ^ { 2 } }$$

Evaluate

$$y$$

Short Solution Steps

Divide \frac{x^{2}+xy-xz}{\left(x+y\right)^{2}-z^{2}} by \frac{x}{\left(x+z\right)^{2}-y^{2}} by multiplying \frac{x^{2}+xy-xz}{\left(x+y\right)^{2}-z^{2}} by the reciprocal of \frac{x}{\left(x+z\right)^{2}-y^{2}}.

$$\frac{\left(x^{2}+xy-xz\right)\left(\left(x+z\right)^{2}-y^{2}\right)}{\left(\left(x+y\right)^{2}-z^{2}\right)x}\times \frac{xy-y^{2}-yz}{\left(x-y\right)^{2}-z^{2}}$$

Factor the expressions that are not already factored in \frac{\left(x^{2}+xy-xz\right)\left(\left(x+z\right)^{2}-y^{2}\right)}{\left(\left(x+y\right)^{2}-z^{2}\right)x}.

$$\frac{x\left(x+y+z\right)\left(x+y-z\right)\left(x-y+z\right)}{x\left(x+y+z\right)\left(x+y-z\right)}\times \frac{xy-y^{2}-yz}{\left(x-y\right)^{2}-z^{2}}$$

Cancel out x\left(x+y+z\right)\left(x+y-z\right) in both numerator and denominator.

$$\left(x-y+z\right)\times \frac{xy-y^{2}-yz}{\left(x-y\right)^{2}-z^{2}}$$

Factor the expressions that are not already factored in \frac{xy-y^{2}-yz}{\left(x-y\right)^{2}-z^{2}}.

$$\left(x-y+z\right)\times \frac{y\left(x-y-z\right)}{\left(x-y+z\right)\left(x-y-z\right)}$$

Cancel out x-y-z in both numerator and denominator.

$$\left(x-y+z\right)\times \frac{y}{x-y+z}$$

Cancel out x-y+z and x-y+z.

$$y$$

Differentiate w.r.t. y

$$1$$

Steps Using Definition of a Derivative

Divide \frac{x^{2}+xy-xz}{\left(x+y\right)^{2}-z^{2}} by \frac{x}{\left(x+z\right)^{2}-y^{2}} by multiplying \frac{x^{2}+xy-xz}{\left(x+y\right)^{2}-z^{2}} by the reciprocal of \frac{x}{\left(x+z\right)^{2}-y^{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\left(x^{2}+xy-xz\right)\left(\left(x+z\right)^{2}-y^{2}\right)}{\left(\left(x+y\right)^{2}-z^{2}\right)x}\times \frac{xy-y^{2}-yz}{\left(x-y\right)^{2}-z^{2}})$$

Factor the expressions that are not already factored in \frac{\left(x^{2}+xy-xz\right)\left(\left(x+z\right)^{2}-y^{2}\right)}{\left(\left(x+y\right)^{2}-z^{2}\right)x}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{x\left(x+y+z\right)\left(x+y-z\right)\left(x-y+z\right)}{x\left(x+y+z\right)\left(x+y-z\right)}\times \frac{xy-y^{2}-yz}{\left(x-y\right)^{2}-z^{2}})$$

Cancel out x\left(x+y+z\right)\left(x+y-z\right) in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(x-y+z\right)\times \frac{xy-y^{2}-yz}{\left(x-y\right)^{2}-z^{2}})$$

Factor the expressions that are not already factored in \frac{xy-y^{2}-yz}{\left(x-y\right)^{2}-z^{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(x-y+z\right)\times \frac{y\left(x-y-z\right)}{\left(x-y+z\right)\left(x-y-z\right)})$$

Cancel out x-y-z in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(x-y+z\right)\times \frac{y}{x-y+z})$$

Cancel out x-y+z and x-y+z.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y)$$

The derivative of ax^{n} is nax^{n-1}.

$$y^{1-1}$$

Subtract 1 from 1.

$$y^{0}$$

For any term t except 0, t^{0}=1.

$$1$$