How do you differentiate # ln(x+4+e^-3x)#?

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Answer 1 · 2018-03-14T13:24:45

#color(blue)((1-3e^(-3x))/(x+4+e^(-3x)))#

If:

#y=ln(x)<=>e^y=x#

Using this definition for the given function:

#e^y=x+4+e^(-3x)#

Differentiating implicitly:

#e^ydy/dx=1+0-3e^(-3x)#

Dividing by: #color(white)(88)bb(e^y)#

#dy/dx=(1-3e^(-3x))/e^y#

From above:

#e^y=x+4+e^(-3x)#

#:.#

#dy/dx=color(blue)((1-3e^(-3x))/(x+4+e^(-3x)))#