If f(x+y)=f(x)+f(y)+5xy and lim f(h)/h =3. h»0
Find f'(x)
To find the derivative of f(x), we need to use the definition of the derivative:
f'(x) = lim(h->0) [f(x+h) - f(x)] / h
In this case, we are given that lim(f(h)/h) = 3 as h approaches 0. This is the definition of the derivative at a point, and it implies that f'(0) = 3.
To find the derivative of f(x) at any other point, we need to find the derivative using the definition.
Let's evaluate the expression [f(x+h) - f(x)] / h using the given equation f(x+y) = f(x) + f(y) + 5xy:
[f(x+h) - f(x)] / h = [f(x) + f(h) + 5xh - f(x)] / h
= [f(h) + 5xh] / h
Now, taking the limit as h approaches 0:
lim(h->0) [f(h) + 5xh] / h
The first term f(h) is a constant with respect to h, so its derivative is 0. The second term 5xh is a linear function of h, so its derivative with respect to h is 5x. Since we are taking the limit as h approaches 0, the value of 5x does not depend on h. Therefore, we can rewrite the expression as:
lim(h->0) [0 + 5xh] / h = lim(h->0) [5xh] / h = lim(h->0) 5x = 5x
Hence, the derivative of f(x) is given by:
f'(x) = 5x