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The derivative of a curve is basically the slope of the curve. If we say, for example, that if y = 2x, the derivative is 2, that means that at any point the line has this slope. If we say that for the function y = x2, the derivative is 2x, that means that at any point "x", the slope is twice the value of "x".
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What does formula of derivative of an inverse allows you to do?
The formula for the derivative of an inverse (finv)' = 1/(f' o (finv)) allows you get a formula for the derivative of the inverse of any function that you already know the derivative of. For example: What is the derivative of sqrt(x)? You could figure this out using the definition of the derivative, but it is complicated. You already know that the derivative of x2 is 2x. So let f = x2; finv = sqrt(x), f' = 2x. This gives: (sqrt(x))' = 1/(2 sqrt(x)). Now you have derived a "square root rule" with almost no work.
Proof of the derivative of the cosecant function?
Express the cosecant in terms of sines and cosines; in this case, csc x = 1 / sin x. This can also be written as (sin x)-1. Remember that the derivative of sin x is cos x, and use either the formula for the derivative of a quotient (using the first expression), or the formula for the derivative of a power (using the second expression).
What is the second derivitive of sec x?
Write sec x as a function of sines and cosines (in this case, sec x = 1 / cos x). Then use the division formula to take the first derivative. Take the derivative of the first derivative to get the second derivative. Reminder: the derivative of sin x is cos x; the derivative of cos x is - sin x.
Derivative of 2aXaXa with respect to x?
If by "2aXaXa", you actually mean "2a3", then the derivative with respect to a is 6a2. On the other hand, if you actually mean "2a3X2", then it's derivative with respect to X would be 6a2X2(da/dx) + 4a3X. If "a" is simply a constant though, then it's derivative is 4a3X
What does the constant rate mean in math?
It means that the first derivative is a constant. The derivative may be with regard to time or any other variable.
