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

##### 2 Answers
Apr 30, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 \ln 10 \cdot {10}^{3 x - 4}$

#### Explanation:

$y = {10}^{3 x - 4}$

$\ln y = \left(3 x - 4\right) \cdot \ln 10$

$\frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = \ln 10 \cdot \frac{d}{\mathrm{dx}} \left(3 x - 4\right)$ [Implicit differentiation and standard differential]

$\frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = \ln 10 \cdot \left(3 - 0\right)$ [Power rule]

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 \ln 10 \cdot y$

$= 3 \ln 10 \cdot {10}^{3 x - 4}$

Apr 30, 2017

A few additional observations

#### Explanation:

Notice that ${10}^{3 x - 4} = {10}^{3 x} / {10}^{4}$
So that ${10}^{3 x - 4} = {1000}^{x} / 10000$
Therefore if $y = {10}^{3 x - 4}$ then $y = {1000}^{x} / 10000$

This gives us $\frac{\mathrm{dy}}{\mathrm{dx}} = \left(\frac{1}{10000}\right) {1000}^{x} \cdot \ln \left(1000\right)$.
If we wish to simplify ln(1000), we have
$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(\frac{3 \ln 10}{10000}\right) {1000}^{x}$.

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