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Topic 8: Differentiation

 1. DEFINITIONS The derivative of a function at point is given by the number This number must be finite. In addition that limit must exist. The left hand derivative of is the number and the right hand derivative is the number . The function is differentiable at point if and only if it has both right hand and left hand derivatives at and they are equal. A function is differentiable on an interval if it is differentiable at every point of that interval. Example FInd derivative of function at Solution We know that Since then is not differentiable at 0. Notation We have used the notation to denote the derivative of the function . There are also many other ways to denote the derivative like . Theorem If a function is differentiable on interval I then it is also continuous on that interval. The converse of this theorem is not true. Example The function is continuous at but not differentiable at 2. DIFFERENTIALS Consider a differentiable function . The differential of this function is denoted and defined by . From this relation we can write . Example If then 3. RULES FOR DIFFERENTIATIONS 3.1. Derivative of a sum If are two differentiable functions, then In fact, 3.2. Multiplication by a scalar Consider a real number k and a differentiable function then . In fact, 3.3. Derivative of a product If are two differentiable functions, then In fact, 3.4. Derivative of the inverse of a function Consider a differentiable function different from zero then In fact, 3.5. Derivative of a quotient Given two differentiable functions then, In fact, 3.6. Derivative of a power Given a differentiable function then In fact, Verify for n = 1 Then, it is true for n=2. Suppose that the property is true for all n, it means Verify that it is true for  The property is true for , then it is true for all n Thus, Particular case   3.7. Derivative of a composite function Given two differentiable functions then, In fact, 3.8. Derivative of a reciprocal function Recall that  4. DERIVATIVE OF TRIGONOMETRIC FUNCTIONS 4.1. Function sine and cosine Function sine and cosine are differentiable on set of real numbers. In addition, In fact, Generally, if u is a differentiable function then Also, recall that  Generally, if u is a differentiable function then 4.2. Function tangent and cotangent The function tangent is differentiable on . In addition, In fact, Generally, if u is a differentiable function then Also, recall that then Generally, if u is a differentiable function then 4.3. Function arcsine and arccosine   4.4. Function arctangent and arccotangent We know that  In the same way, Generally, if u is a differentiable function then 5. SUCCESSIVE DERIVATIVES Successive derivatives of a function are higher order derivatives of the same function. The second derivative of function y = f  is The third derivative of function y = f  is The fourth derivative of function y = f  is The nth derivative of function y = f  is 6. APPLICATIONS 6.1. Critical points is a critical point of the functions if exists and if or Example Find all critical points of the function Solution The critical points are 6.2. Maximumu and minimum values To find the maximum and the minimum values in given interval, follow the following steps: find all critical point in the given interval evaluate the function at the critical points found and at the end points of the given interval identify the absolute the maximum and minimum values. Example Find the maximum and minimum values of the following functions Solution Now, From this list Maximum value is Minimum values are 6.3. Increasing and decreasing of a function   6.4. Concavity of a function   6.5. L'Hospital rule