This will require the application of the chain rule--twice.
First, for #e^(5x^2 + x + 3)#
Let #y = e^u#, and #u = 5x^2 + x + 3#
#dy/dx = dy/(du) xx (du)/dx#
The derivative of #e^u# is #e^u#. The derivative of #5x^2 + x + 3# is #10x + 1#.
Hence,
#dy/dx = e^u xx 10x + 1#
#dy/dx = (10x + 1)e^(5x^2 + x + 3)#
Now for the second application of the chain rule.
Let #y = sqrt(u) = u^(1/2)# and #u = e^(5x^2 + x + 3)#
We already know the derivative of #u#. The derivative of #y#, by the power rule, is #1/2u^(-1/2) = 1/(2u^(1/2))#.
Hence,
#dy/dx = 1/(2u^(1/2)) xx (10x + 1)e^(5x^2 + x + 3)#
#dy/dx = ((10x + 1)e^(5x^2 + x + 3))/(2sqrt(e^(5x^2 + x + 3))#
#f'(x) = ((10x + 1)e^(5x^2 + x + 3))/(2sqrt(e^(5x^2 + x + 3))#
#f'(x) = ((10x + 1)e^(5x^2 + x + 3))/(2(e^(5x^2 + x + 3))^(1/2)#
By the quotient rule of exponents: #a^n/a^m = a^(n - m)#:
#f'(x) = ((10x + 1)sqrt(e^(5x^2 + x + 3)))/2#
Hopefully this helps!
