can someone please prove the difference rule to me?

d/dx[f(x)-g(x)]=d/dx f(x)- d/dx g(x)

Go to the definition of a derivative:

Lim x>0 of {f(x+deltax)-f(x)]/deltaX

To prove the difference rule, we need to start from the definition of a derivative. The derivative of a function f(x) represents its rate of change at a particular point.

The definition of the derivative of a function f(x) is:

Lim Δx→0 of [f(x + Δx) - f(x)] / Δx

Now, let's consider the difference between two functions, f(x) and g(x), and take their derivative using the definition:

Lim Δx→0 of [f(x + Δx) - g(x + Δx) - (f(x) - g(x))] / Δx

Expanding the terms inside the limit:

Lim Δx→0 of [f(x + Δx) - g(x + Δx) - f(x) + g(x)] / Δx

Now, we can separate the limit into three limits using algebraic property:

Lim Δx→0 of [f(x + Δx) - f(x)] / Δx + Lim Δx→0 of [g(x) - g(x + Δx)] / Δx - Lim Δx→0 of [f(x) - g(x)] / Δx

Simplifying further:

Lim Δx→0 of [f(x + Δx) - f(x)] / Δx - Lim Δx→0 of [g(x + Δx) - g(x)] / Δx - Lim Δx→0 of [f(x) - g(x)] / Δx

Using the definition of the derivative, we can rewrite the three limits as the derivatives of f(x), g(x), and -g(x):

d/dx f(x) - d/dx g(x) - d/dx (-g(x))

Since the derivative of -g(x) is the negative of the derivative of g(x), we can simplify further:

d/dx f(x) - d/dx g(x) + d/dx g(x)

Finally, combining like terms:

d/dx f(x) - d/dx g(x)

Therefore, we have proved the difference rule:

d/dx [f(x) - g(x)] = d/dx f(x) - d/dx g(x)