Let f be a function and a be the given point you are considering. Then,
f(x) - f(a)
---------------
(x-a)
is the difference quotient. If the limit as x approaches a exists, then the function is differentiable at a, or we say the derivative exists at a. If that limit does not exist, then the derivative does not exist at that point.
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If the second derivative of a function is zero, then the function has a constant slope, and that function is linear. Therefore, any point that belongs to that function lies on a line.
The answer will depend on the context. If the curve in question is a differentiable function then the gradient of the tangent is given by the derivative of the function. The gradient of the tangent at a given point can be evaluated by substituting the coordinate of the point and the equation of the tangent, though that point, is then given by the point-slope equation.
A function is differentiable at a point if the derivative exists there.
The graph of a linear function is a line with a constant slope. The graph of an exponential function is a curve with a non-constant slope. The slope of a given curve at a specified point is the derivative evaluated at that point.
the deivative of a function is the gradient, at a point if you can sub in the x coordinate for that point