Contents
Can two different functions have the same derivatives?
The meaning is quite clear. If two functions have the same derivative, then the tangent lines at corresponding points should have the same slopes, or to put it another way, their graphs should go up and down in the same way. If the two graphs also share a point and they are parallel, then they must be the same.
Can a function have a family of derivatives?
In particular, just as we can creat first and second derivative sign charts for a single function, we can often do so for entire families of functions where critical numbers and possible inflection points depend on arbitrary constants.
How do you differentiate a function with two variables?
In implicit differentiation, we differentiate each side of an equation with two variables (usually x and y) by treating one of the variables as a function of the other. This calls for using the chain rule. Let’s differentiate x 2 + y 2 = 1 x^2+y^2=1 x2+y2=1x, squared, plus, y, squared, equals, 1 for example.
Can two functions have the same gradient?
If two functions have the same gradient, then they are the same function.
Do equal functions have equal derivatives?
The derivative of two equal functions is indeed equal.
How many derivatives can you take on a function?
In other words, when you differentiate, you don’t get two derivatives for one function, rather two derivatives corresponding to two different functions, one y=41/55×1/5+1×3/4, and the other, y=41/55×1/5−1×3/4.
Can a function have more than one Antiderivative?
Every function that has at least one antiderivative, has more than one antiderivative. More precisely, it has an infinite number of antiderivatives. The difference between two antiderivatives is a constant.
Can two different functions have the same Taylor series?
Yes. It is possible that two functions could have the same Taylor series around some point but differ elsewhere, for example the functions y=0 and the function exp(-1/x2) for all x not zero, y=0 at x=0. The two functions have the same Taylor series at x=0, but not elsewhere.
Is the derivative unique?
Anti-derivatives “reverse differentiation”, but unlike derivatives, they are not unique. Instead, anti-derivatives are unique up to adding a constant.