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

1 Answer
Sep 22, 2015

Answer:

#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)#