Question

determine $h(x)$ if $h(x)=log_5(4x^{3}+3x^{2}+6x)$. select the correct answer below: $\frac{(ln 5)(12x^{2}+6x + 6)}{4x^{3}+3x^{2}+6x}$, $log_5(12x^{2}+6x + 6)$, $\frac{12x^{2}+6x + 6}{4x^{3}+3x^{2}+6x}$, $\frac{1}{(ln 5)(4x^{3}+3x^{2}+6x)}$, $\frac{12x^{2}+6x + 6}{(ln 5)(4x^{3}+3x^{2}+6x)}$

Explanation:

Step1: Use log - derivative formula

The derivative of $\log_a(u)=\frac{1}{u\ln a}\cdot u'$, where $u = 4x^{3}+3x^{2}+6x$ and $a = 5$.

Step2: Differentiate $u$

$u'=(4x^{3}+3x^{2}+6x)'=12x^{2}+6x + 6$.

Step3: Calculate $h'(x)$

$h'(x)=\frac{12x^{2}+6x + 6}{(\ln 5)(4x^{3}+3x^{2}+6x)}$.

Answer:

$\frac{12x^{2}+6x + 6}{(\ln 5)(4x^{3}+3x^{2}+6x)}$ (last option)