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User:JaviPrieto/Derivatives

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(Click for larger image) At each point, the derivative of is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where green, negative where red, and zero where black.

Q(h) is the slope of the secant line between (a, ç'(a)) and (a + h, ç'(a + h)). If ç' is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from the point h = 0. If the limit exists, meaning that there is a way of choosing a value for Q(0) that makes the graph of Q a continuous function, then the function ç' is differentiable at the point a, and its derivative at a equals Q(0).

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  or  
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  or   ,

wherever this function is defined. For example, if , then

for all functions ƒ and g and all real numbers a and b.
for all functions ƒ and g.
for all functions ƒ and g where g ? 0.
  • Chain rule: If , then

Up to changing variables, this is the statement that the function is the best linear approximation to ç' at a.