How do you find the derivative of y=ln(70x^2+24x+7)?

Jun 25, 2018

$y ' = \frac{140 x + 24}{70 {x}^{2} + 24 x + 7}$

Explanation:

Note that

$\left(\ln \left(x\right)\right) ' = \frac{1}{x}$
We also Need the chain rule. So we get

$y ' = \frac{140 x + 24}{70 {x}^{2} + 24 x + 7}$

Jun 25, 2018

$f ' \left(y\right) = \frac{140 x + 24}{70 {x}^{2} + 24 x + 7}$

Explanation:

Bit old fashioned in my approach so I will be using format type $\frac{\mathrm{dy}}{\mathrm{dx}}$

Set $f \left(x\right) = t = 70 {x}^{2} + 24 x + 7 \textcolor{w h i t e}{\text{d}} \to \frac{\mathrm{dt}}{\mathrm{dx}} = 140 x + 24$

Set $y = \ln \left(t\right) \to \frac{\mathrm{dy}}{\mathrm{dt}} = \frac{1}{t}$

Combining these as: $\frac{\mathrm{dy}}{\mathrm{dt}} \times \frac{\mathrm{dt}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{dx}}$

So applying the above:

$\frac{\mathrm{dy}}{\mathrm{dx}} = f ' \left(x\right) = \textcolor{w h i t e}{\text{d}} \frac{1}{70 {x}^{2} + 24 x + 7} \times \left(140 x + 24\right)$

$\textcolor{w h i t e}{\text{dddd.ddddd") = color(white)("d}} \frac{140 x + 24}{70 {x}^{2} + 24 x + 7}$