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Question Number 44190 by olj55336@awsoo.com last updated on 23/Sep/18

$$$$$$$$$$$$$$$$$$$$$$Q..Findsecondderivative..$$$$5^{x}solvepleasestapbystap$$

Commented by maxmathsup by imad last updated on 23/Sep/18

$$letf\left(x\right)=a^{x}witha>0anda\ne 1letfindgenerally{f}^{\left(n\right)}\left(x\right)$$$$wehavef\left(x\right)=e^{xln\left(a\right)}\Rightarrow f^{\left(1\right)}\left(x\right)=ln\left(a\right)e^{xln\left(a\right)}\Rightarrow f^{\left(2\right)}\left(x\right)={\left\{ln\left(a\right)\right\}}^2{e}^{xln\left(a\right)}$$$$letsuppose{f}^{\left(n\right)}\left(x\right)={\left\{ln\left(a\right)\right\}}^n{e}^{xln\left(a\right)}\Rightarrow f^{(n+1)}\left(x\right)={\left\{ln\left(a\right)\right\}}^{n+1}{e}^{xln\left(a\right)}$$$$so{f}^{\left(n\right)}\left(x\right)={\left\{ln\left(a\right)\right\}}^n{a}^x$$$$forf\left(x\right)=5^x\Rightarrow f^{\left(2\right)}\left(x\right)={\left\{ln\left(5\right)\right\}}^2{5}^x.$$

Answered by MrW3 last updated on 23/Sep/18

$$y=5^x=e^{x\ln 5}$$$$\frac{dy}{dx}=\left(\ln 5\right)e^{x\ln 5}$$$$\frac{d^{2}y}{{dx}^2}={\left(\ln 5\right)}^2{e}^{x\ln 5}={\left(\ln 5\right)}^2{5}^x$$

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