can someone please prove the difference rule to me?
d/dx[f(x)-g(x)]=d/dx f(x)- d/dx g(x)
plz help
Of course! I'll be happy to explain and prove the difference rule for derivatives. The difference rule states that the derivative of the difference between two functions is equal to the difference of their derivatives.
To prove this, let's consider two differentiable functions, f(x) and g(x). We'll start by finding the derivative of the difference between the two functions, which is:
d/dx [f(x) - g(x)]
To calculate this derivative, we can apply the rules of differentiation. The derivative of a difference is the difference of the derivatives. So, we can find the derivatives of f(x) and g(x) separately:
d/dx f(x) - d/dx g(x)
Now, let's analyze each part separately. The derivative of f(x) with respect to x is denoted as d/dx f(x) and it represents the rate of change of f(x) with respect to x. Similarly, we have d/dx g(x) which represents the rate of change of g(x) with respect to x.
Since f(x) and g(x) are both differentiable functions, their derivatives exist. By subtracting d/dx g(x) from d/dx f(x), we are essentially subtracting the rate of change of g(x) from the rate of change of f(x). This result therefore represents the rate of change of the difference between the two functions.
Hence, we have proven that:
d/dx [f(x) - g(x)] = d/dx f(x) - d/dx g(x)
I hope this explanation helps clarify the proof of the difference rule for you!