How would I find the derivative of x^2 + 3x using the definition of the derivative?
d/dx x^2+3x = 2x+3
the derivative of a function f(x) is
Limit [f(x+h) - f(x)]/h
h--> 0
so derivative of x^2 + 3x =
Lim[(x+h)^2 + 3(x+h) - x^2 - 3x]/h =
h--> 0
Lim [2hx + h^2 + 3h]/h =
h--> 0
Lim [2x + h + 3\]
h--> 0
= 2x + 3
To find the derivative of a function using the definition of the derivative, you need to apply the limit definition of the derivative.
The definition of the derivative of a function f(x) at a point x=a is given by:
f'(a) = lim(h->0) [f(a+h) - f(a)] / h
In this case, the function is f(x) = x^2 + 3x.
To find the derivative, we need to evaluate the above expression by substituting f(x) into it.
f'(a) = lim(h->0) [(a+h)^2 + 3(a+h) - (a^2 + 3a)] / h
Now, simplify the expression by expanding and collecting like terms:
f'(a) = lim(h->0) [(a^2 + 2ah + h^2 + 3a + 3h) - (a^2 + 3a)] / h
Simplify further:
f'(a) = lim(h->0) [2ah + h^2 + 3h] / h
Now, cancel out the common factor of h in the numerator:
f'(a) = lim(h->0) [2a + h + 3]
Taking the limit as h approaches 0, we can evaluate the expression:
f'(a) = 2a + 3
Therefore, the derivative of f(x) = x^2 + 3x is f'(x) = 2x + 3.