Differentiation

Question

A curve has the equation y=3e^2x.

Find the value of d²y/dx² at the point where the gradient is 3.

 

Ans:6

 

Thank You! :)

Answer 1

y = 3e^2x

 

Use the chain rule to find the first derivative.  Let u = 2x:

 

y' = dy/dx = dy/du * du/x = d(3e^u)/du * d(2x)/dx = 6e^2x

 

Apply the chain rule again to get the second derivative:

 

y'' = d²y/dx² = 12e^2x

 

Gradient = first derivative (y'=3).  Solve for x.

 

3 = 6e^2x

1/2 = e^2x

ln(1/2) = 2x

 

(1/2)ln(1/2) = x

 

To find the value of y'' (second derivative) when y'=3, plug x = (1/2)ln(1/2) into the second derivative equation:

 

y'' = 12e^{2(1/2)ln(1/2)}

y'' = 12e^ln(1/2)

 

y'' = 12*(1/2) = 6