Solve f(a+h)-f(a)=f(a+h)-f(a)/h*h

$$f ( a + h ) - f ( a ) = \frac { f ( a + h ) - f ( a ) } { h } \cdot h$$

Solve for a (complex solution)

$$a\in \mathrm{C}$$

$$h\neq 0$$

Solution Steps

Multiply both sides of the equation by h.

$$f\left(a+h\right)h-fah=\left(f\left(a+h\right)-fa\right)h$$

Use the distributive property to multiply f by a+h.

$$\left(fa+fh\right)h-fah=\left(f\left(a+h\right)-fa\right)h$$

Use the distributive property to multiply fa+fh by h.

$$fah+fh^{2}-fah=\left(f\left(a+h\right)-fa\right)h$$

Combine fah and -fah to get 0.

$$fh^{2}=\left(f\left(a+h\right)-fa\right)h$$

Use the distributive property to multiply f by a+h.

$$fh^{2}=\left(fa+fh-fa\right)h$$

Combine fa and -fa to get 0.

$$fh^{2}=fhh$$

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

$$fh^{2}=fh^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{C}$$

Solve for f (complex solution)

$$f\in \mathrm{C}$$

$$h\neq 0$$

Solution Steps

Multiply both sides of the equation by h.

$$f\left(a+h\right)h-fah=\left(f\left(a+h\right)-fa\right)h$$

Use the distributive property to multiply f by a+h.

$$\left(fa+fh\right)h-fah=\left(f\left(a+h\right)-fa\right)h$$

Use the distributive property to multiply fa+fh by h.

$$fah+fh^{2}-fah=\left(f\left(a+h\right)-fa\right)h$$

Combine fah and -fah to get 0.

$$fh^{2}=\left(f\left(a+h\right)-fa\right)h$$

Use the distributive property to multiply f by a+h.

$$fh^{2}=\left(fa+fh-fa\right)h$$

Combine fa and -fa to get 0.

$$fh^{2}=fhh$$

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

$$fh^{2}=fh^{2}$$

Subtract fh^{2} from both sides.

$$fh^{2}-fh^{2}=0$$

Combine fh^{2} and -fh^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any f.

$$f\in \mathrm{C}$$

Solve for a

$$a\in \mathrm{R}$$

$$h\neq 0$$

Solution Steps

Multiply both sides of the equation by h.

$$f\left(a+h\right)h-fah=\left(f\left(a+h\right)-fa\right)h$$

Use the distributive property to multiply f by a+h.

$$\left(fa+fh\right)h-fah=\left(f\left(a+h\right)-fa\right)h$$

Use the distributive property to multiply fa+fh by h.

$$fah+fh^{2}-fah=\left(f\left(a+h\right)-fa\right)h$$

Combine fah and -fah to get 0.

$$fh^{2}=\left(f\left(a+h\right)-fa\right)h$$

Use the distributive property to multiply f by a+h.

$$fh^{2}=\left(fa+fh-fa\right)h$$

Combine fa and -fa to get 0.

$$fh^{2}=fhh$$

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

$$fh^{2}=fh^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{R}$$

Solve for f

$$f\in \mathrm{R}$$

$$h\neq 0$$

Solution Steps

Multiply both sides of the equation by h.

$$f\left(a+h\right)h-fah=\left(f\left(a+h\right)-fa\right)h$$

Use the distributive property to multiply f by a+h.

$$\left(fa+fh\right)h-fah=\left(f\left(a+h\right)-fa\right)h$$

Use the distributive property to multiply fa+fh by h.

$$fah+fh^{2}-fah=\left(f\left(a+h\right)-fa\right)h$$

Combine fah and -fah to get 0.

$$fh^{2}=\left(f\left(a+h\right)-fa\right)h$$

Use the distributive property to multiply f by a+h.

$$fh^{2}=\left(fa+fh-fa\right)h$$

Combine fa and -fa to get 0.

$$fh^{2}=fhh$$

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

$$fh^{2}=fh^{2}$$

Subtract fh^{2} from both sides.

$$fh^{2}-fh^{2}=0$$

Combine fh^{2} and -fh^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any f.

$$f\in \mathrm{R}$$