The answer to taylor 3^x is 1+ln(3)x+(ln^2(3))/2 x^2+(ln^3(3))/6 x^3+(ln^4(3))/(24)x^4+\ldots

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taylor 3^x

Solution

taylor

3^{x}

1 + ln (3) x + \frac{ln ^{2} ( 3 )}{2} x^{2} + \frac{ln ^{3} ( 3 )}{6} x^{3} + \frac{ln ^{4} ( 3 )}{24} x^{4} + \dots

Solution steps

Apply the Taylor Formula

= 1 + \frac{\frac{d}{d x} ( 3 ^{x} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( 3 ^{x} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( 3 ^{x} ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= 1 + \frac{ln ( 3 )}{1 !} x + \frac{ln ^{2} ( 3 )}{2 !} x^{2} + \frac{ln ^{3} ( 3 )}{3 !} x^{3} + \frac{ln ^{4} ( 3 )}{4 !} x^{4} + \dots

Refine

= 1 + ln (3) x + \frac{ln ^{2} ( 3 )}{2} x^{2} + \frac{ln ^{3} ( 3 )}{6} x^{3} + \frac{ln ^{4} ( 3 )}{24} x^{4} + \dots

x \to 1 lim (- 4 x + 6)

\int 19 z^{3} e^{z} d z

\frac{d y}{d t} = y - 3 e^{- t}

\frac{d y}{d x} = e^{2 x + 3 y}

What is taylor 3^x ?
The answer to taylor 3^x is 1+ln(3)x+(ln^2(3))/2 x^2+(ln^3(3))/6 x^3+(ln^4(3))/(24)x^4+\ldots