Solve deltay=x^3-(x+deltax)^3/(x+deltax)^3(x)^3

$$\delta y = \frac { x ^ { 3 } - ( x + \delta x ) ^ { 3 } } { ( x + \delta x ) ^ { 3 } ( x ) ^ { 3 } }$$

Solve for y

$$\left\{\begin{matrix}y=-\frac{\delta ^{2}+3\delta +3}{\left(x\left(\delta +1\right)\right)^{3}}\text{, }&\delta \neq -1\text{ and }x\neq 0\\y\in \mathrm{R}\text{, }&x\neq 0\text{ and }\delta =0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(\delta +1\right)^{3}x^{6}.

$$\delta y\left(\delta +1\right)^{3}x^{6}=x^{3}-\left(x+\delta x\right)^{3}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(\delta +1\right)^{3}.

$$\delta y\left(\delta ^{3}+3\delta ^{2}+3\delta +1\right)x^{6}=x^{3}-\left(x+\delta x\right)^{3}$$

Use the distributive property to multiply \delta y by \delta ^{3}+3\delta ^{2}+3\delta +1.

$$\left(y\delta ^{4}+3y\delta ^{3}+3y\delta ^{2}+\delta y\right)x^{6}=x^{3}-\left(x+\delta x\right)^{3}$$

Use the distributive property to multiply y\delta ^{4}+3y\delta ^{3}+3y\delta ^{2}+\delta y by x^{6}.

$$y\delta ^{4}x^{6}+3y\delta ^{3}x^{6}+3y\delta ^{2}x^{6}+\delta yx^{6}=x^{3}-\left(x+\delta x\right)^{3}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+\delta x\right)^{3}.

$$y\delta ^{4}x^{6}+3y\delta ^{3}x^{6}+3y\delta ^{2}x^{6}+\delta yx^{6}=x^{3}-\left(x^{3}+3x^{2}\delta x+3x\delta ^{2}x^{2}+\delta ^{3}x^{3}\right)$$

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

$$y\delta ^{4}x^{6}+3y\delta ^{3}x^{6}+3y\delta ^{2}x^{6}+\delta yx^{6}=x^{3}-\left(x^{3}+3x^{3}\delta +3x\delta ^{2}x^{2}+\delta ^{3}x^{3}\right)$$

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

$$y\delta ^{4}x^{6}+3y\delta ^{3}x^{6}+3y\delta ^{2}x^{6}+\delta yx^{6}=x^{3}-\left(x^{3}+3x^{3}\delta +3x^{3}\delta ^{2}+\delta ^{3}x^{3}\right)$$

To find the opposite of x^{3}+3x^{3}\delta +3x^{3}\delta ^{2}+\delta ^{3}x^{3}, find the opposite of each term.

$$y\delta ^{4}x^{6}+3y\delta ^{3}x^{6}+3y\delta ^{2}x^{6}+\delta yx^{6}=x^{3}-x^{3}-3x^{3}\delta -3x^{3}\delta ^{2}-\delta ^{3}x^{3}$$

Combine x^{3} and -x^{3} to get 0.

$$y\delta ^{4}x^{6}+3y\delta ^{3}x^{6}+3y\delta ^{2}x^{6}+\delta yx^{6}=-3x^{3}\delta -3x^{3}\delta ^{2}-\delta ^{3}x^{3}$$

Combine all terms containing y.

$$\left(\delta ^{4}x^{6}+3\delta ^{3}x^{6}+3\delta ^{2}x^{6}+\delta x^{6}\right)y=-3x^{3}\delta -3x^{3}\delta ^{2}-\delta ^{3}x^{3}$$

The equation is in standard form.

$$\left(\delta ^{4}x^{6}+3\delta ^{3}x^{6}+3\delta ^{2}x^{6}+\delta x^{6}\right)y=-x^{3}\delta ^{3}-3\delta ^{2}x^{3}-3\delta x^{3}$$

Divide both sides by \delta ^{4}x^{6}+3\delta ^{3}x^{6}+3\delta ^{2}x^{6}+\delta x^{6}.

$$\frac{\left(\delta ^{4}x^{6}+3\delta ^{3}x^{6}+3\delta ^{2}x^{6}+\delta x^{6}\right)y}{\delta ^{4}x^{6}+3\delta ^{3}x^{6}+3\delta ^{2}x^{6}+\delta x^{6}}=-\frac{\delta \left(\delta ^{2}+3\delta +3\right)x^{3}}{\delta ^{4}x^{6}+3\delta ^{3}x^{6}+3\delta ^{2}x^{6}+\delta x^{6}}$$

Dividing by \delta ^{4}x^{6}+3\delta ^{3}x^{6}+3\delta ^{2}x^{6}+\delta x^{6} undoes the multiplication by \delta ^{4}x^{6}+3\delta ^{3}x^{6}+3\delta ^{2}x^{6}+\delta x^{6}.

$$y=-\frac{\delta \left(\delta ^{2}+3\delta +3\right)x^{3}}{\delta ^{4}x^{6}+3\delta ^{3}x^{6}+3\delta ^{2}x^{6}+\delta x^{6}}$$

Divide -\delta \left(3+3\delta +\delta ^{2}\right)x^{3} by \delta ^{4}x^{6}+3\delta ^{3}x^{6}+3\delta ^{2}x^{6}+\delta x^{6}.

$$y=-\frac{\delta ^{2}+3\delta +3}{\left(x\left(\delta +1\right)\right)^{3}}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt[3]{\frac{\delta ^{2}+3\delta +3}{y}}}{\delta +1}\text{, }&\delta \neq -1\text{ and }y\neq 0\\x\neq 0\text{, }&\delta =0\end{matrix}\right.$$