If I have a function f(x), and am given its derivative, f'(x): may I take it as a given that f(x) is an integral of f'(x).
My reasoning is that 'undoing' the derivative gives me the derivative.
Eg is 1/3x^3 an integral of x^2?
Thanks.
Yes to both your answers and your reasoning. An arbitrary constant can always be added to the integral, however.
"My reasoning is that 'undoing' the derivative gives me the derivative."
oops, meant to write
......gives me the integral.
Thanks- that was quick!
Therefore, given my scenario, I could use the integral to calculate the area under the curve described by the derivative?
Charlie.
That is what I assumed you meant; I should have read your reasoning more closely. Anyway, you got it right
Great, and thank you very much.
Charlie.
Yes, your reasoning is correct. If you are given the derivative of a function, you can find its original function by integrating the derivative.
In calculus, the process of finding the original function from its derivative is called antidifferentiation or integration.
To check if 1/3x^3 is an integral of x^2, you can differentiate 1/3x^3 and see if it equals x^2.
Let's find the derivative of 1/3x^3:
Using the power rule for differentiation, the derivative of x^n is nx^(n-1).
So, the derivative of 1/3x^3 is (3/3)x^(3-1) = x^2.
Since the derivative of 1/3x^3 equals x^2, we can conclude that 1/3x^3 is indeed an integral of x^2.
So, in general, if you have a function f(x) and its derivative f'(x), you can take f(x) as an integral of f'(x) because integration is the process of finding the original function from its derivative.