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Exponential Function and Derivative of Exponential Function
- 1. CALCULAS DERIVATIVE OF EXPONETS HAFIZ MUHAMMAD ATTA UR REHMAN 0028 GROUP NO.1 UNIVERERSITY OF NAROWAL
- 2. CONTENTS ● WHAT IS EXPONENTIAL FUNCTION? ● DERIVATIVE OF EXPONENTIAL FUNCTION
- 3. EXPONENTIAL FUNCTION f ( x ) = a x < O R > y = a x , a > 1 Example y=10x A mathematical function in which an independent variable appears in one of the exponents is called exponential function.
- 4. Derivative of Exponential Function General Form 𝑑 𝑑𝑥 𝑒 𝑥 = 𝑒 𝑥 𝑑 𝑑𝑥 𝑥 Mnemonics At first the exponential function comes as it is,then we take the derivative of the power.𝑑 𝑑𝑥 𝑥 =1 𝐴𝑠 𝑤𝑒 𝑘𝑛𝑜𝑤 𝑑 𝑑𝑥 𝑒 𝑥 = 𝑒 𝑥
- 5. Examples Power Rule𝑑 𝑑𝑥 𝑒 𝑥2 = 𝑒 𝑥2 𝑑 𝑑𝑥 𝑥2 𝐴𝑐𝑐𝑜𝑟𝑑𝑖𝑛𝑔 𝑡𝑜 𝑃𝑜𝑤𝑒𝑟 𝑅𝑢𝑙𝑒 𝑑 𝑑𝑥 𝑒 𝑥 = 𝑒 𝑥2 2𝑥 𝑑 𝑑𝑥 𝑥 𝑛 = 𝑛𝑥 𝑛−1 𝑑 𝑑𝑥 𝑥3 = 3𝑥3−1 𝑥3 = 3𝑥2 𝑑 𝑑𝑥 𝑒 𝑥2 = 𝑒 𝑥2 𝑑 𝑑𝑥 𝑥2 = 2𝑥 𝑒 𝑥2 𝑑 𝑑𝑥 𝑒 𝑥 = 𝑒 𝑥2 2𝑥2−1
- 6. Examples 𝑑 𝑑𝑥 𝑒 𝑥2+1 𝑑 𝑑𝑥 𝑒 𝑥2+1 = 𝑒 𝑥2+1 2𝑥 Derivative of x2=2x Derivative of 1=0
- 7. Examples Product Rule 𝑥𝑒 𝑥 𝐴𝑐𝑐𝑜𝑟𝑑𝑖𝑛𝑔 𝑡𝑜 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑅𝑢𝑙𝑒 = 𝑒𝑥(𝑥 + 1) 𝑓 𝑥 . 𝑔(𝑥) =f(x)g’x +g(x)f’(x) = 𝑥𝑒 𝑥 +𝑒 𝑥
- 8. Examples Mnemonics 𝑒 𝑎𝑥 𝑠𝑖𝑛𝑏𝑥 =𝑒 𝑎𝑥 𝑑 𝑑𝑥 𝑠𝑖𝑛𝑏𝑥+ 𝑠𝑖𝑛𝑏𝑥 𝑑 𝑑𝑥 𝑒 𝑎𝑥 According to Product Rule First will come as it is, derivative of Second plus second as it is and derivative of first =𝑒 𝑎𝑥 𝑑 𝑑𝑥 𝑐𝑜𝑠𝑏𝑥(𝑏) Remainder Derivative of sin=cos + 𝑠𝑖𝑛𝑏𝑥 𝑒 𝑎𝑥(a)
- 10. THANK YOU!