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Continuity And Differentiability

Question

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Differentiate e^x + e^x² +....+ e^x⁵. w.r.t.x.

Solution

Let space space space space space space space space straight y equals straight e to the power of straight x plus straight e to the power of straight x squared end exponent plus straight e to the power of straight x cubed end exponent plus straight e to the power of straight x to the power of 4 plus straight e to the power of straight x to the power of 5 therefore space dy over dx equals straight d over dx open parentheses straight e to the power of straight x plus straight e to the power of straight x squared end exponent plus straight e to the power of straight x cubed end exponent plus straight e to the power of straight x to the power of 4 plus straight e to the power of straight x to the power of 5 close parentheses space space space space space space space space space space space space space equals straight d over dx open parentheses straight e to the power of straight x close parentheses plus straight d over dx open parentheses straight e to the power of straight x squared end exponent close parentheses plus straight d over dx open parentheses straight e to the power of straight x cubed end exponent close parentheses plus straight d over dx open parentheses straight e to the power of straight x to the power of 4 close parentheses plus straight d over dx open parentheses straight e to the power of straight x to the power of 5 close parentheses space space space space space space space space space space space space space equals straight e to the power of straight x. straight d over dx left parenthesis straight x right parenthesis plus straight e to the power of straight x squared end exponent. straight d over dx left parenthesis straight x squared right parenthesis plus straight e to the power of straight x cubed end exponent. straight d over dx left parenthesis straight x cubed right parenthesis plus straight e to the power of straight x to the power of 4. straight d over dx left parenthesis straight x to the power of 4 right parenthesis plus straight e to the power of straight x to the power of 5 straight d over dx open parentheses straight x to the power of 5 close parentheses space space space space space space space space space space space space space equals straight e to the power of straight x.1 plus straight e to the power of straight x squared end exponent.2 straight x plus straight e to the power of straight x cubed end exponent.3 straight x squared plus straight e to the power of straight x to the power of 4.4 straight x cubed plus straight e to the power of straight x to the power of 5.5 straight x to the power of 4 space space space space space space space space space space space space space equals straight e to the power of straight x plus 2 straight x space straight e to the power of straight x squared end exponent plus 3 straight x squared straight e to the power of straight x cubed end exponent plus 4 straight x cubed straight e to the power of straight x to the power of 4 plus 5 straight x to the power of 4 straight e to the power of straight x to the power of 5

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