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

1 Answer

Answer:

#color(blue)(f' (x)=(-(3x+5)*sqrt(1-(3x+5)^2)*e^(-sqrt(1-(3x+5)^2)))/(3x^2+10x+8))#

Explanation:

The given equation is #f(x)=1/e^(sqrt(1-(3x+5)^2))#

The solution:

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

#f' (x)=d/dx(1/e^(sqrt(1-(3x+5)^2)))=d/dx(e^(-sqrt(1-(3x+5)^2)))#

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

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

#f' (x)=#

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

#f' (x)=#

#e^(-sqrt(1-(3x+5)^2))*(-1/(2sqrt(1-(3x+5)^2)))(0-2(3x+5)^1*3)#

#f' (x)=#

#e^(-sqrt(1-(3x+5)^2))*(-1/(2sqrt(1-(3x+5)^2)))(-6(3x+5))#

#f' (x)=e^(-sqrt(1-(3x+5)^2))*((9x+15)/(sqrt(1-(3x+5)^2)))#

We simplify by rationalization

#f' (x)=(-(9x+15)*sqrt(1-(3x+5)^2)*e^(-sqrt(1-(3x+5)^2)))/(9x^2+30x+24)#

Reducing the numbers by canceling common numerical coefficients

#color(blue)(f' (x)=(-(3x+5)*sqrt(1-(3x+5)^2)*e^(-sqrt(1-(3x+5)^2)))/(3x^2+10x+8))#

God bless....I hope the explanation is useful.