How do you differentiate #f(x)=e^sqrt(1-(3x+5)^2)# using the chain rule.?

1 Answer
Jan 7, 2016

Answer:

#dy/dx = -(3e^(sqrt(1-(3x+5)^2))*(3x+5))/(sqrt(1-(3x+5)^2)#

Explanation:

#y = e^(sqrt(1-(3x+5)^2))#

Say that #sqrt(1-(3x+5)^2) = u#

#dy/dx = d/dxe^(u)(du)/dx#
#dy/dx = e^(u)d/dxsqrt(1-(3x+5)^2)#

Say that #v =1-(3x+5)^2 #

#dy/dx = e^(u)d/dxsqrt(v)(dv)/dx#
#dy/dx = e^(u)/(2sqrt(v))d/dx(1-(3x+5)^2)#

Say that #3x + 5 = w#

#dy/dx = e^(u)/(2sqrt(v))d/dx(1-w^2)(dw)/dx#
#dy/dx = -(e^(u)w)/(sqrt(v))d/dx(3x+5)#
#dy/dx = -(3e^(u)w)/(sqrt(v))#

Now put it all back in terms of #x#

#dy/dx = -(3e^(sqrt(1-(3x+5)^2))*(3x+5))/(sqrt(1-(3x+5)^2)#