The derivative of a continuous function at x is the slope of the tangent line to the curve at x. The attached pdf file develops the idea of a derivative first using slopes of secant lines and then introducing and explaining the difference quotient in detail. An example and an explanation are provided for using the limit of the

Verify that the solution of u"=f(x), u(0)=0, u(1)=0 given by u(x)= the integration from 0 to 1 of k(x,y)f(y)dy. Use Leibniz rule.
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Please see the attached file for the fully formatted problems.
The Chain Rule states how to differentiate composite functions
If and be differentiable, then the derivative of the composite function is
In Leibnitz notation, if and , then .
For example,
or
The derivative of the exponential function is

1.
The sets and are defined as follows.
Write and using interval notation.
If the set is empty, write .
2.
Graph the system below and write its solution.
Note that you can also answer "No solution" or "Infinitely many solutions."
3.
Solve the compound inequality.
or
Write the solution in

Need help finding partial derivatives of attached problems.
Find three partial derivatives of the function r with respect to x, y, and z
1. r = uvw − u^2 − v^2 − w^2 where u = y + z, v = x + z,w = x + y
2. r = p / q + q / s + s / p where p = e^yz, q = e^xz, s = e^xy

1. Find the derivatives for the following functions ("^" means "to the power of", sorry I can't do double exponents on my keyboard) :
a. f(X) = 100e10X
b. f(X) = e(10X-5)
c. f(X) = e^X3
d. f(X) = 2X2e^(1- X2)
e. f(X) = 5Xe(12- 2X)
f. f(X) = 100e^(X3 + X4)
g. f(X) = e^(200X - X2 + X100)
2. Fi

Please see the attachment. I am trying to take the first, second, and third derivatives of this equation, so I can utilize them to determine relationship to force and modulus, and thermal expansion coefficient. Please help me take the derivatives properly. Thanks!

Radical and rational exponent notation are two ways to show the same process.
Explain the similarities between radicals and rational exponent notation.
Provide at least two other examples of mathematical notation or wording denoting the same process.