If #f(x)= 2 x^2 + x # and #g(x) = 2e^x + 1 #, how do you differentiate #f(g(x)) # using the chain rule?

2 Answers
Jan 3, 2018

Answer:

#16e^(2x)+10e^x#

Explanation:

#f(g(x)#

#=f(2e^x+1)=2(2e^x+1)^2+2e^x+1#

#color(white)(xxxxxxxxx)=2(4e^(2x)+4e^x+1)+2e^x+1#

#color(white)(xxxxxxxxx)=8e^(2x)+8e^x+2+2e^x+1#

#color(white)(xxxxxxxxx)=8e^(2x)+10e^x+3#

#rArrd/dx(8e^(2x)+10e^x+3)#

#=8e^(2x)xxd/dx(2x)+10e^xlarrcolor(blue)"chain rule"#

#=16e^(2x)+10e^x#

Jan 3, 2018

Answer:

See below.

Explanation:

#f(x)=2x^2+x# , #g(x)=2e^x+1#

#:.#

#f(g(x))=2(2e^x+1)^2+2e^x+1=8e^(2x)+10e^x+3#

Using the chain rule:

#dy/dx=(dy)/(du)*(du)/dx#

Let #u=e^x#

#8u^2+10u+3#

#dy/(dx)(8u^2)=dy/(du)(8u^2) * (du)/(dx)(u)=16u*e^x=16e^(2x)#

#dy/(dx)(10u)=dy/(du)(10u) * (du)/(dx)(u)=10 * e^x=10e^x#

#:.#

#dy/dx(8x^(2x)+10e^x+3)=16e^(2x)+10e^x#