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

Q . . F i n d s e c o n d d e r i v a t i v e . .

5^{x} s o l v e p l e a s e s t a p b y s t a p

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

l e t f (x) = a^{x} w i t h a > 0 a n d a \neq 1 l e t f i n d g e n e r a l l y f^{(n)} (x)

w e h a v e f (x) = e^{x l n (a)} \Rightarrow f^{(1)} (x) = l n (a) e^{x l n (a)} \Rightarrow f^{(2)} (x) = {l n (a)}^{2} e^{x l n (a)}

l e t s u p p o s e f^{(n)} (x) = {l n (a)}^{n} e^{x l n (a)} \Rightarrow f^{(n + 1)} (x) = {l n (a)}^{n + 1} e^{x l n (a)}

s o f^{(n)} (x) = {l n (a)}^{n} a^{x}

f o r f (x) = 5^{x} \Rightarrow f^{(2)} (x) = {l n (5)}^{2} 5^{x} .

Answered by MrW3 last updated on 23/Sep/18

y = 5^{x} = e^{x \ln 5}

\frac{d y}{d x} = (\ln 5) e^{x \ln 5}

\frac{d^{2} y}{{d x}^{2}} = {(\ln 5)}^{2} e^{x \ln 5} = {(\ln 5)}^{2} 5^{x}

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