Calculus Section 3.1 Calculate the derivative of a function using the limit definition Recall: The slope of a line is given by the formula m = y 2 – y.

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Presentation transcript:

Calculus Section 3.1 Calculate the derivative of a function using the limit definition Recall: The slope of a line is given by the formula m = y 2 – y 1 x 2 – x 1 The slope of a line is constant. The slope of a curve is variable. The slope of a curve is the slope of a line tangent to the curve at a given point.

Consider a point P on the graph of f(x). Let there be another point Q on the graph of f(x). The secant line containing P and Q has slope that is slightly different from the slope of the line tangent to the curve at point p. The slope of the line PQ is given by m = f(x+∆x) – f(x) x+∆x - x Let the coordinates of P be (x,f(x)) and the horizontal distance from P to Q be ∆x. The coordinates of Q are then (x+ ∆x, f(x+ ∆x)).

Let point Q approach point P. The slope of line PQ approaches the slope of the curve at point P. The value of ∆x approaches zero. The derivative of a function is a formula that gives the slope of the function at any point on the graph of the function. The slope of the tangent line at point P is given by: This formula is called the derivative of f(x).

The notation for the derivative of the function f(x) is: f’(x) y’ f(x) D x y D x f(x)

Find the derivative of the function. f(x) = x 2

Find the derivative of the function. Find the slope of f(x) at the point (1,0). f(x) = 2x 2 + 3x – 5

Find the derivative of the function. Find the slope of f(x) at the point (-2,-24) y = x 3 + 4x - 8

Find the derivative of the function. f(x) =

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