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)