What is taylor 5^x,0 ?

The answer to taylor 5^x,0 is 1+ln(5)x+(ln^2(5))/2 x^2+(ln^3(5))/6 x^3+(ln^4(5))/(24)x^4+\ldots

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taylor 5^x,0

taylor

5^{x}, 0

1 + ln (5) x + \frac{ln ^{2} ( 5 )}{2} x^{2} + \frac{ln ^{3} ( 5 )}{6} x^{3} + \frac{ln ^{4} ( 5 )}{24} x^{4} + \dots

Solution steps

Apply the Taylor Formula

= 1 + \frac{\frac{d}{d x} ( 5 ^{x} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( 5 ^{x} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( 5 ^{x} ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= 1 + \frac{ln ( 5 )}{1 !} x + \frac{ln ^{2} ( 5 )}{2 !} x^{2} + \frac{ln ^{3} ( 5 )}{3 !} x^{3} + \frac{ln ^{4} ( 5 )}{4 !} x^{4} + \dots

Refine

= 1 + ln (5) x + \frac{ln ^{2} ( 5 )}{2} x^{2} + \frac{ln ^{3} ( 5 )}{6} x^{3} + \frac{ln ^{4} ( 5 )}{24} x^{4} + \dots

y = (x^{4} + 2)^{2} (x^{5} + 4)^{4}

\frac{d}{d x} (e^{4} x^{7})

\int (3 x^{2} - 2)^{2} d x

f (x) = x^{2} + 5

y^{^{'}} = - \frac{9 x}{16 y}

What is taylor 5^x,0 ?
The answer to taylor 5^x,0 is 1+ln(5)x+(ln^2(5))/2 x^2+(ln^3(5))/6 x^3+(ln^4(5))/(24)x^4+\ldots