# Derivative of Integral of Continuous Function f(x)

• kidia
In summary, the derivative of a function f(x) that is continuous over an interval containing (a,x) can be found using first principles by taking the derivative of the antiderivative of f(x) over that interval. This can be simplified to f(x) using the fundamental theorem of calculus. If this is for AP calculus, it is recommended to review the fundamental theorem of calculus.
kidia
If f is continuous over an interval containing (a,x)find from first principles the derivative of the function f(x)=integral f(t)dt.Any help?

dump rule

integral from a->x of f(t)dt = f(x) * d/dx(x) = f(x)

you can derive this easily
let F(x) be the antiderivative of f(x)

therefore the integral = F(x) - F(a)

take the derivative of that... the F(a) term falls off cause its a constant

thus you do d/dx(F(x)) = f(x)

yay!

if this is for AP calc, i would really recommend looking over the fundamental theorom of calculus

I don't think that's as "first principles" as kidia intended. Here is the standard proof of the fundamental theorem:
Let $$F(x)= \int_a^x f(t)dt$$. Then $$F(x+h)= \int_a^{x+h}f(t)dt$$
$$= \int_a^x f(t)dt+ \int_x^{x+h}f(t)dt$$

So that F(x+h)- F(x)= \int_x^{x+h}f(t)dt. Now apply the mean value theorem to the function $$\int_x^{x+h}f(t)dt$$ to argue that F(x+h)-F(x)= hf(x*) where x* is between x and x+h. Finally, divide both sides by h and take the limit as h goes to 0.

## 1. What is the derivative of the integral of a continuous function f(x)?

The derivative of the integral of a continuous function f(x) is the original function f(x). This is known as the fundamental theorem of calculus. It states that the derivative of the integral of a function is equal to the original function.

## 2. Why is the derivative of the integral of a continuous function f(x) equal to the original function?

This is because the derivative of a function measures the rate of change of that function at a given point. The integral, on the other hand, measures the area under the curve of the function. These two operations are inverses of each other, meaning they undo each other's effects. Therefore, the derivative of the integral is the original function.

## 3. Does this property hold for all continuous functions?

Yes, this property holds for all continuous functions. As long as the function is continuous and has a defined derivative, the fundamental theorem of calculus applies.

## 4. Are there any other important applications of this property?

Yes, the fundamental theorem of calculus has many important applications in mathematics and science. It is used in physics to calculate displacement, velocity, and acceleration, and in economics to calculate marginal cost, revenue, and profit. It also plays a crucial role in the field of differential equations.

## 5. How is this property related to the area under a curve?

The fundamental theorem of calculus connects the concept of area under a curve to the concept of slope. The integral of a function represents the area under the curve, and the derivative of the integral gives the slope of the curve. This relationship allows us to use integration to solve problems involving area and slope, making it a powerful tool in many fields.

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