The Definition of the DerivativeSlope of the Tangent LineDerivative and DifferentiabilityDeri vatives GraphicallyDifferentiability and ContinuityDerivatives as Functions
2 The Definition of the Derivative Definition 3 The Definition of the Derivative Definition Notations Recall that by previous considerations the derivative of a function f at the point x0 is the slope of the tangent line of the graph of f at the point (x0f(x0)). This is an important geometric view to the derivative. 4 Slope of the Tangent Line (1) To compute the equation of the tangent line at a point (af(a)) of the graph of a function f observe that 5 Slope of the Tangent Line (2) Example 6 Slope of the Tangent Line (2) Example Solution 7 Slope of the Tangent Line (2) Example Solution 8 Slope of the Tangent Line (2) Example Solution 9 Slope of the Tangent Line (2) Example Solution 10 Slope of the Tangent Line (3) Example Solution (contd) Conclusion 11 Derivative and Differentiability (1) Definition 1 12 Derivative and Differentiability (1) Definition 1 Definition 2 13 Derivative and Differentiability (1) Definition 1 Definition 2 14 Derivative and Differentiability (2) Theorem 15 Derivative and Differentiability (2) Theorem Proof
16 Derivative and Differentiability (3) Theorem Proof (contd)
17 Differentiability Graphically 18 Differentiability Graphically 19 Differentiability Graphically 20 Differentiability Graphically 21 Differentiability and Continuity Theorem Proof 22 The Derivative as a Function Definition Notation The derivative of a function f is a function whose the domain of definition consists of all the numbers x such that the function f has its derivative at the point x. 23 Graphs of Functions and their Derivatives Problem The picture on the right shows the graphs of two functions f and g and the graphs of their derivatives. The maximum of the shown values of f is larger than that of g. Which is which 24 Graphs of Functions and their Derivatives Df Problem f The picture on the right shows the graphs of two functions f and g and the graphs of their derivatives. The maximum of the shown values of f is larger than that of g. Which is which Solution The red curve appears to intersect the x-axis whenever the tangent of the blue curve is horizontal. Hence the red curve is the graph of the derivative of the function f. The blue curve is the graph of f. g Dg The green curve is everywhere decreasing. Hence its tangents have negative slopes. The black curve is the only curve that takes only negative values. Hence the green curve is the graph of the function g and the black curve that of its derivative.
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