Derivative of a definite integral?

In summary, the conversation discusses the concept of the derivative of a definite integral and whether it is always equal to zero. It is clarified that the derivative of a definite integral is not always zero, and the Leibniz Rule must be applied to find the derivative. It is also mentioned that the function being integrated may not necessarily be differentiable.
  • #1
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consider x is between the interval [a,b]
would it be correct to say that the derivative of a definite integral F(x) is f(x) because as dx approaches zero in (x + dx), the width of ALL "imaginary rectangles" would closely resemble a line segment which approximates f(x)? therefore change in area under a curve is dependent to the change in the height of f(x) with respect to dx(which is inifinitesimally small)??

the different notations used in several videos i watched seemed to have confused me or doubt my own understanding of a seemingly simple concept
 
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  • #2
Derivative of a definite integral? The definite integral calculates an orientated area. This is a constant. The derivative of a constant equals zero. Therefor, the derivative of a definite integral is zero.
 
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  • #3
Math_QED said:
Derivative of a definite integral? The definite integral calculates an orientated area. This is a constant. The derivative of a constant equals zero. Therefor, the derivative of a definite integral is zero.
sorry. just integral, not definite integral
 
  • #4
Math_QED said:
Derivative of a definite integral? The definite integral calculates an orientated area. This is a constant. The derivative of a constant equals zero. Therefor, the derivative of a definite integral is zero.
Not necessarily.

You have to apply the Leibniz Rule:

https://en.wikipedia.org/wiki/Leibniz_integral_rule
 
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  • #5
Math_QED said:
Derivative of a definite integral? The definite integral calculates an orientated area. This is a constant. The derivative of a constant equals zero. Therefor, the derivative of a definite integral is zero.

Sorry. I made a mistake. If we simply discard the limit then what happens? That is if $$F(x)=\int f(x)dx$$ implies $$F'(x)=\int f'(x)dx$$?
 
  • #6
Ahmed Mehedi said:
Sorry. I made a mistake. If we simply discard the limit then what happens? That is if $$F(x)=\int f(x)dx$$ implies $$F'(x)=\int f'(x)dx$$?
No, by definition ##\int f(x) dx## is a function ##F(x)## satisfying ##F'(x) = f(x)##. So rather ##F'(x) = f(x)##.

What you wrote is not entirely false, but then you have to assume that ##f## is differentiable which must not be the case. Also you must take into account annoying (integrating) constants.
 
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  • #7
Math_QED said:
No, by definition ##\int f(x) dx## is a function ##F(x)## satisfying ##F'(x) = f(x)##. So rather ##F'(x) = f(x)##.

What you wrote is not entirely false, but then you have to assume that ##f## is differentiable which must not be the case. Also you must take into account annoying (integrating) constants.

Can't we just differentiate both sides of the first line and pass the differentiation operator inside the integration operator?
 
  • #8
Math_QED said:
No, by definition ##\int f(x) dx## is a function ##F(x)## satisfying ##F'(x) = f(x)##. So rather ##F'(x) = f(x)##.

What you wrote is not entirely false, but then you have to assume that ##f## is differentiable which must not be the case. Also you must take into account annoying (integrating) constants.

I got your answer. But why you assume that f is not differentiable?
 
  • #9
Ahmed Mehedi said:
I got your answer. But why you assume that f is not differentiable?

##F## can be a primitive function of ##f## without ##f## being differentiable. For example, take ##f(x)=|x|##. This is not differentiable in ##0## but by the fundamental theorem of calculus there exists a primitive ##F## for ##f##.
 
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1. What is the definition of the derivative of a definite integral?

The derivative of a definite integral is the rate of change of the integral with respect to its upper limit. It represents the slope of the tangent line to the area under the curve at a specific point.

2. How do you calculate the derivative of a definite integral?

The derivative of a definite integral can be calculated by first evaluating the integral and then taking the derivative of the resulting function.

3. What is the relationship between the derivative of a definite integral and the original function?

The derivative of a definite integral is the same as the original function evaluated at the upper limit of integration. In other words, the derivative of a definite integral is a function of the upper limit.

4. Can the derivative of a definite integral be negative?

Yes, the derivative of a definite integral can be negative. This would indicate that the area under the curve is decreasing as the upper limit of integration increases.

5. Why is the derivative of a definite integral important?

The derivative of a definite integral has many applications in physics, engineering, and economics. It can be used to calculate rates of change, velocities, and other important quantities in various real-world scenarios.

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