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)