Question

What is the first differential of `y=10^(3x-4)` ?

Answer 1

`dy/dx= 3ln10*10^(3x-4)`

Explanation:

`y=10^(3x-4)`

`lny = (3x-4)*ln10`

`1/y dy/dx = ln10*d/dx(3x-4)` [Implicit differentiation and standard differential]

`1/y dy/dx= ln10*(3-0)` [Power rule]

`dy/dx = 3ln10*y`

`= 3ln10*10^(3x-4)`

Answer 2

A few additional observations

Explanation:

Notice that `10^(3x-4) = 10^(3x)/10^4`
So that `10^(3x-4) = 1000^(x)/10000`
Therefore if `y = 10^(3x-4)` then `y = 1000^(x)/10000`

This gives us `dy/dx = (1/10000)1000^x*ln(1000)`.
If we wish to simplify ln(1000), we have
`dy/dx = ((3ln10)/10000)1000^x`.

This answer is equal to the one given in the original response.