Showing the properties of differentiating an integral

• greswd
In summary, we discussed how to show that the derivative of the integral of y with respect to x is equal to y times the derivative of x with respect to t. This can be proven using the chain rule by considering y as a function of x and x as a function of t. This is not a homework problem, but rather a question out of curiosity. Textbooks may not have the exact answers, but they typically follow a similar template for solving these types of problems.
greswd
How do we show that

$$\frac{d}{dt}\left[\int\!y\,\mathrm{d} x\right] = y\,\frac{dx}{dt}$$

greswd said:
How do we show that

$$\frac{d}{dt}\left[\int\!y\,\mathrm{d} x\right] = y\,\frac{dx}{dt}$$
Is this a homework problem?

greswd said:
How do we show that

$$\frac{d}{dt}\left[\int\!y\,\mathrm{d} x\right] = y\,\frac{dx}{dt}$$
Are we to assume, here, that y and x are functions of t? If we assume that y is a function of x only (with no "t" that is not in the "x") and x is a function of t, then we an write y(x(t)).

Of course, then $$F(x)= \int y dt$$ is the function such that dF/dx= y. Given that, we have that $d/dt(\int y dx)= dF/dt= (dF/dx)(dx/dt)= y(x)(dx/dt)$ by the chain rule.

Mark44 said:
Is this a homework problem?

Nope. Homework questions are usually standard, and answers are all in the textbooks.
I came up with this problem just out of curiosity.

Anyway, thanks for the solution HallsofIvy

Last edited:
greswd said:
Nope. Homework questions are usually standard, and answers are all in the textbooks.

DrewD said:

1. What is the purpose of showing the properties of differentiating an integral?

The purpose of showing the properties of differentiating an integral is to demonstrate the relationship between differentiation and integration. This helps to understand and solve more complex mathematical problems involving integrals.

2. What are the main properties of differentiating an integral?

The main properties of differentiating an integral include the power rule, the constant multiple rule, the sum rule, and the product rule. These rules allow for the differentiation of different types of integrals.

3. How do the properties of differentiating an integral differ from those of regular differentiation?

The properties of differentiating an integral differ from those of regular differentiation because integrals involve a sum of infinitely small values, whereas regular differentiation involves finding the slope of a single function at a specific point. This requires the use of different rules and techniques.

4. How do the properties of differentiating an integral help in solving real-world problems?

The properties of differentiating an integral can help in solving real-world problems by allowing us to find rates of change, areas under curves, and volumes of irregular shapes. This has applications in physics, engineering, economics, and other fields.

5. Are there any limitations or exceptions to the properties of differentiating an integral?

Yes, there are some limitations and exceptions to the properties of differentiating an integral. For example, the chain rule does not apply to integrals, and some integrals may require the use of more advanced techniques such as integration by parts or substitution. It is important to understand these limitations and exceptions in order to effectively apply the properties of differentiating an integral.

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