Solve f^prime(x)=ax^2+bx+L

$$f ^ { \prime } ( x ) = a x ^ { 2 } + b x + L$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{bx+L}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&L=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$ax^{2}+bx+L=\frac{\mathrm{d}}{\mathrm{d}x}(f)x$$

Subtract bx from both sides.

$$ax^{2}+L=\frac{\mathrm{d}}{\mathrm{d}x}(f)x-bx$$

Subtract L from both sides.

$$ax^{2}=\frac{\mathrm{d}}{\mathrm{d}x}(f)x-bx-L$$

The equation is in standard form.

$$x^{2}a=-bx-L$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=\frac{-bx-L}{x^{2}}$$

Dividing by x^{2} undoes the multiplication by x^{2}.

$$a=\frac{-bx-L}{x^{2}}$$

Divide -bx-L by x^{2}.

$$a=-\frac{bx+L}{x^{2}}$$

Solve for L

$$L=-x\left(ax+b\right)$$