Differential of #z=3^sin(3x+7y)# ?

1 Answer
Steve M
Mar 9, 2018

# z_x = (partial z)/(partial x) = ln 3 \ 3^(sin(3x+7y)+1) \ cos(3x+7y) #

# z_y = (partial z)/(partial y) = 7 \ ln 3 \ 3^sin(3x+7y) \ cos(3x+7y) #

Explanation:

We have:

# z = 3^sin(3x+7y) #

Using the result:

# d/dx a^x = lna \ a^x #

Along with the chain rule, we can readily compute the partial derivatives:

# z_x = (partial)/(partial x) 3^sin(3x+7y)#
# \ \ \ = ln 3 \ 3^sin(3x+7y) \ (partial)/(partial x) sin(3x+7y) #
# \ \ \ = ln 3 \ 3^sin(3x+7y) \ cos(3x+7y) \ (partial)/(partial x) (3x+7y)#
# \ \ \ = ln 3 \ 3^sin(3x+7y) \ cos(3x+7y) \ 3#
# \ \ \ = ln 3 \ 3^(sin(3x+7y)+1) \ cos(3x+7y) #

Similarly:

# z_y = (partial)/(partial y) 3^sin(3x+7y)#
# \ \ \ = ln 3 \ 3^sin(3x+7y) \ cos(3x+7y) \ 7#
# \ \ \ = 7 \ ln 3 \ 3^sin(3x+7y) \ cos(3x+7y) #

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