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Is there any other anti derivative of 1 divided by x?
The antiderivative of 1/x is ln(x) + C. That is, to the natural (base-e) logarithm, you can add any constant, and still have an antiderivative. For example, ln(x) + 5. These are the only antiderivatives; there are no different functions that have the same derivatives. This is valid, in general, for all antiderivatives: if you have one antiderivative of a function, all other antiderivatives are obtained by adding a constant.
Why does an answer to an integration problem involve a Constant of Integration?
The indefinite integral is the anti-derivative - so the question is, "What function has this given function as a derivative". And if you add a constant to a function, the derivative of the function doesn't change. Thus, for example, if the derivative is y' = 2x, the original function might be y = x squared. However, any function of the form y = x squared + c (for any constant c) also has the SAME derivative (2x in this case). Therefore, to completely specify all possible solutions, this constant should be added.
What is the purpose of differential calculus?
Differential Calculus is to take the derivative of the function. It is important as it can be applied and supports other branches of science. For ex, If you have a velocity function, you can get its acceleration function by taking its derivative, same relationship as well with area and volume formulas.
When finding the derivative of a point on a piecewise function does every function in the piecewise function need to be continuous and approach the same limit?
All differentiable functions need be continuous at least.
What is the integral of the derivative with respect to x of a function of x divided by that same function of x with respect to x?
∫ f'(x)/f(x) dx = ln(f(x)) + C C is the constant of integration.
