The sum of the derivative and integral of a constant equals what? Explain in great detail along with mathematical expressions

Answer 1

Let's start by considering a constant function, f(x) = c, where c is a constant. The derivative of a constant function is always zero, because the value of the function does not change as x varies.

Mathematically, we have:

f(x) = c

f'(x) = 0 (since the derivative of a constant is zero)

Now, let's find the integral of f(x) over an interval [a, b]. The integral of a constant function is computed by multiplying the constant by the width of the interval.

∫[a, b] f(x) dx = ∫[a, b] c dx = c∫[a, b] dx

Integrating dx simply adds the variable x, evaluated at the limits of integration a and b:

∫[a, b] dx = (x)|[a, b] = (b - a)

Therefore:

∫[a, b] f(x) dx = c∫[a, b] dx = c(b - a)

Now, let's sum the derivative and integral of the constant function:

f'(x) + ∫[a, b] f(x) dx = 0 + c(b - a) = c(b - a)

So, the sum of the derivative and integral of a constant function f(x) = c over an interval [a, b] is equal to c times the width of the interval (b - a).

In simpler terms, the derivative of a constant function is always zero, and the integral of a constant function is equal to the constant times the width of the interval. Hence, when you sum them up, you get the same result as the integral, which is c times the width of the interval.