Question

How do you use the product Rule to find the derivative of `g(t) = (3^t) - (4^t3)`?

Answer 1

`ln3(3^t) - 3ln4(4^t)`

Explanation:

We'll have to use a little more than the product rule to find this derivative. Luckily, we can take it one step at a time - finding `(dg)/dt3^t` and then `(dg)/dt(4^t*3)`.

`(dg)/dt3^t` is easy - recall that `d/dxa^x = lna(a^x)`. That means `(dg)/dt3^t = ln3(3^t)`.

`(dg)/dt(4^t*3)` is a little more complicated, but bearable. Here's where we utilize the product rule. Remember, the product rule states that `d/dxuv = u'v+uv'`. In our case, `u = 4^t` and `v = 3`. Substituting,

`(dg)/dt(4^t*3) = (4^t)'(3)+(4^t)(3)'`
`(dg)/dt(4^t*3) = 3ln4(4^t)+(4^t)(0)`
`(dg)/dt(4^t*3) = 3ln4(4^t)`

Finally, we can now put it all together:

`(dg)/dt(3^t+4^t*3) = (dg)/dt3^t-(dg)/dt(4^t*3)`
`(dg)/dt(3^t+4^t*3) = ln3(3^t) - 3ln4(4^t)`