Suppose that g(X)=f(cx). Show from the definition that g'(x)=cf'(cx)
To show that g'(x) = cf'(cx), we need to use the definition of the derivative.
Let's start by finding the derivative of g(x) = f(cx). The derivative of a function f(x) with respect to x is denoted as f'(x) and it is defined as the limit of the difference quotient as the change in x approaches zero. Mathematically, it can be expressed as:
f'(x) = lim(h->0) [(f(x + h) - f(x))/h]
Now, let's proceed to find the derivative of g(x) = f(cx). We substitute cx for x in the above difference quotient:
g'(x) = lim(h->0) [f(cx + h) - f(cx))/h]
Next, we can factor out c from the terms inside the brackets and apply the definition of derivative to f(cx):
g'(x) = lim(h->0) [f(cx + h) - f(cx))/h] * c
Now, we can rewrite the terms inside the brackets using the definition of f'(cx):
g'(x) = lim(h->0) [f'(cx) * (cx + h - cx))/h] * c
Simplifying the expression inside the brackets:
g'(x) = lim(h->0) [f'(cx) * h/h] * c
The h terms cancel out, leaving us with:
g'(x) = lim(h->0) [f'(cx)] * c
Since the limit of a constant multiplied by a function is equal to the constant times the limit of the function, we have:
g'(x) = [f'(cx)] * c
Therefore, we have proven that g'(x) = cf'(cx) using the definition of the derivative.