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

2 Answers
Feb 25, 2016

Answer:

#f'(x) = (10x + 1 )e^(5x^2+x+3)#

Explanation:

Using the #color(blue)" chain rule "#

#d/dx[f(g(x)) ] = f'(g(x)).g'(x) #

and #d/dx(e^x) = e^x#

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

# = e^(5x^2+x+3)(10x + 1 )#

Feb 25, 2016

Answer:

#h'(x)=e^(5x^2+x+3)(10x+1)#

Explanation:

The given equation is #h(x)=e^(5x^2+x+3)#. But it seems as if it is a function inside a function. So, let's write #g(x)=5x^2+x+3# and #f(x)=e^x#.
So that means #f(g(x))=e^(g(x))=e^(5x^2+x+3#

From chain rule, we have #d/dx(f(g(x))=h'(x)=f'(g(x))* g'(x)#.
So, taking for #f(g(x))# and #g(x)#, we get
#h'(x)=e^(g(x))*d/dx(g(x))==e^g(x)(10x+1)#

You can substitute it all to get back the proper answer.