differentiate $(2x^3 + 3x)(x − 2)(x + 4)$ So, I'm really stuck on this problem.
Differentiate $(2x^3 + 3x)(x − 2)(x + 4)$
This is what I come up with $10x^4+16x^3+39x^2+6x-18$.
But, the answer in the book has $16x^4$ as the leading term
Here's my work:
$(2x^3+3x)d/dx(x^2+2x-8)+(x^2+2x-8)d/dx(2x^3+3x)$
$(2x^3+3x)(2x+2)+(x^2+2x-8)(6x^2+3)$
$(4x^4+4x^3+6x^2+6x+6x^4+12x^3-48x^2+3x^2+6x-24)$
$=10x^4+16x^3+39x^2+6x-18$

Let $y=(2x^3+3x)(x-2)(x+4)$. Then if we are not too fussy about whether this makes sense in the reals, we get
$$\ln y=\ln(2x^3+3x)+\ln(x-2)+\ln(x+4).$$
Differentiating, we get
$$\frac{1}{y}\frac{dy}{dx}=\frac{6x^2+3x}{2x^3+3x}+\frac{1}{x-2}+\frac{1}{x+4},$$
and now we know $\frac{dy}{dx}$.
In this problem, the implicit differentiation approach does not give a significicant advantage over the plain Product Rule. However, it does become useful if the problem is a little more complicated.
