# Find the second derivative of z = 4e^5t with respect to t. Which is the correct option from the following?Find the second derivative of z = 4e^5t with respect to t. Which is the correct...

Find the second derivative of *z = 4e^5t* with respect to *t*. Which is the correct option from the following?

Find the second derivative of *z = 4e^5t* with respect to *t*. Which is the correct option from the following?

*d^2z/dt^2 = 100e^5t*

*d^2z/dt^2 = 4e^5t*

*d^2z/dt^2 = 20e^4t*

*d^2z/dt^2 = 25e^5t*

*d^2z/dt^2 = 20e^5t*

*d^2z/dt^2 = 80e^3t*

*print*Print*list*Cite

You need to differentiate the given function `z(t) = 4e^(5t)` with respect to t, using the chain rule, such that:

`z'(t) = 4*e^(5t)*(5t)'`

`z'(t) =4*5*e^(5t)`

`z'(t) =20*e^(5t)`

You need to differentiate again the result of the first differentiation such that:

`z''(t) = (20*e^(5t))' =gt z''(t) = 20*e^(5t)*(5t)'`

`z''(t) = 20*5*e^(5t)`

`z''(t) = 100e^(5t)`

**Hence, evaluating the second derivative of the function yields `z''(t) = 100e^(5t),` hence, you need to select the first option from the given list.**

The function `z = 4e^(5t)` . The second derivative of z with respect to t can be determined as follows:

`(dz)/(dt) = 4*5*e^(5t) = 20*e^(5t)`

`(d^2z)/(dt^2) = 20*5*e^(5t) = 100*e^(5t)`

**The correct option for the second derivative of `z = 4e^(5t)` with respect to t is `(d^2z)/(dt^2) = 100*e^(5t)` **

the correct answer is first one

*d^2z/dt^2 = 100e^5t*

*Explanation:*

*We have a relation: d(e^at) *

* ---------- = a*e^at*

* dt*

*thus,*

* d(4e^5t) *

* ---------- = 4*5*e^5t=20e^5t-----------A*

* dt*

*take the derivative of answer in A*

* d(20e^5t) *

* ---------- = 20*5*e^5t=100e^5t*

* dt*