What is the second derivative of the absolute function $\left|\frac{x+1}{x+2}\right|$?

This is how I deal with absolute functions:

$$ \begin{align} \left|\frac{x+1}{x+2}\right|&=\sqrt{\left(\frac{x+1}{x+2}\right)^{2}}\\ \\ \frac{d}{dx} \left|\frac{x+1}{x+2}\right|&=\frac{d}{dx} \sqrt{\left(\frac{x+1}{x+2}\right)^{2}}\\ &=\frac{1}{2 \sqrt{\left(\frac{x+1}{x+2}\right)^{2}}}\cdot 2 \left(\frac{x+1}{x+2}\right)\cdot\frac{1}{\left(x+2\right)^{2}}\\ &=\frac{x+1}{\left(x+2\right)^{3}\cdot\left|\frac{x+1}{x+2}\right|} \end{align} $$

Notice the chain rule when I differentiate the square root

Tags:

Calculus

Derivatives

Soft Question

Related

Five roots of $x^5+x+1=0$ and the value of $\prod_{k=1}^{5} (2+x_k^2)$ Proving $(a^2 + 1)(b ^2 + 1)(c ^2 + 1) ≥ 2(ab + bc + ca)$ where $a,b,c$ are real numbers. Geometric proof that a Pythagorean triple has a number divisible by $3$? Replacing improper integral by sum of integrals Proof of $\lim_{n \to \infty}(1+\frac{1}{n})^n=e$ Working with infinitesimals of the form d(f(x)), for example d(ax), and relating them to dx (integration, delta function) Is that possible to partition $(\Bbb R,+)$ into 4 additively closed subsets? Prove that $\triangle ABC=\left(\triangle DEF \cdot \triangle XYZ\right)^{1/2}$ Intuition behind inversions of triangles in circles leading to similar triangles Weierstrass Factorization Theorem, infinite polynomial/infinite power series The derivative $\frac{\mathrm d}{\mathrm dx} x^x=x^x\left(\ln x+1\right)$ is problematic for $x<0$ How many subsets of $\{1,2,...,n\}$ do not contain three consecutive integers?

Recent Posts