What is the derivative of #e^x sin(3^(1/2x))#?

1 Answer
Steve M
May 2, 2018

# d/dx \ e^x sin(3^(1/2x)) = ( (ln3 \ 3^(x/2) \ cos(3^(x/2)))/2 + sin(3^(x/2)))e^x \ #

Explanation:

We seek:

# d/dx \ e^x sin(3^(1/2x)) #

Using the product rule we have:

# d/dx \ e^x sin(3^(1/2x)) = e^x (d/dx sin(3^(1/2x))) + (d/dx e^x)sin(3^(1/2x)) #

By utilizing the property #a^x -= e^(xlna)#, and applying the chain rule, we have:

# d/dx sin(3^(1/2x)) = cos(3^(1/2x)) \ d/dx (3^(1/2x))#
# " "= cos(3^(1/2x)) \ d/dx (e^(1/2xln3))#
# " "= cos(3^(1/2x)) \ e^(1/2xln3) \ d/dx (1/2xln3)#
# " "= cos(3^(1/2x)) \ 3^(1/2x) \ (1/2ln3)#

Thus:

# d/dx \ e^x sin(3^(1/2x)) = e^x \ cos(3^(1/2x)) \ 3^(1/2x) \ (1/2ln3) + e^x \ sin(3^(1/2x)) #

# " " = ( (ln3 \ 3^(x/2) \ cos(3^(x/2)))/2 + sin(3^(x/2)))e^x \ #

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