We'll begin by getting the first derivative.
#[1]" "d/dx(4e^(2x)secx)#
The first property we need is #d/dx[cf(x)]=cd/dx[f(x)]#. This means we can factor the constant out of the derivative (which in this case, is #4#).
#[2]" "=4d/dx(e^(2x)secx)#
Next, we need the product rule: #d/dx[f(x)g(x)=f'(x)g(x)+f(x)g'(x)]#. In this case, let's have our #f(x)=e^(2x)# and our #g(x)=secx#.
#[3]" "=4[(d/dxe^(2x))(secx)+(e^(2x))(d/dxsecx)]#
The derivative of #e^x# is just #e^x#. But since we have #e^(2x)#, we need to apply the chain rule. So #d/dx(e^(2x))=2e^(2x)#.
#[4]" "=4[(2e^(2x))(secx)+(e^(2x))(d/dxsecx)]#
The derivative of #secx# is #secxtanx#.
#[5]" "=4[(2e^(2x))(secx)+(e^(2x))(secxtanx)]#
Simplifying, we get:
#[6]" "=color(blue)(8e^(2x)secx+4e^(2x)secxtanx=4e^(2x)secx(2+tanx))#
To get the second derivative, we'll get the derivative of the first derivative.
#[1]" "d/dx[4e^(2x)secx(2+tanx)]#
It's useful to note that #4e^(2x)secx# is the same as our original equation.
Now we apply product rule.
#[2]" "=[d/dx(4e^(2x)secx)][2+tanx]+(4e^(2x)secx)[d/dx(2+tanx)]#
From earlier, we know that #d/dx(4e^(2x)secx)=4e^(2x)secx(2+tanx)#. So we just plug that in.
#[3]" "=[4e^(2x)secx(2+tanx)][2+tanx]+(4e^(2x)secx)[d/dx(2+tanx)]#
The derivative of any constant is #0#, and the derivative of #tanx# is #sec^2x#. So the derivative of #2+tanx# is just #sec^2x#.
#[4]" "=[4e^(2x)secx(2+tanx)][2+tanx]+[4e^(2x)secx][sec^2x]#
We just simplify this now.
#[5]" "=color(red)(4e^(2x)(2+tanx)^2secx+4e^(2x)sec^3x)#
