Second derivative

Differentiate $f(x)$ once to get $f'(x)$, then differentiate that to get $f''(x)$. It is also written $y''$ or $\frac{d^2y}{dx^2}$.

Differentiate one step at a time and write out $f'(x)$ along the way; this keeps coefficients from getting mixed up. The sign of $f''(x)$ is used later to study concavity.

Examples

1. 01What is f''(x)?$\displaystyle f(x)=x^3$AnswerHide

   Answer$\displaystyle 6x$

   Steps

   1. Differentiate once
   2. $\displaystyle \frac{d}{dx}\,x^{3}$$\displaystyle = 3x^{2}$
   3. Differentiate the result again
   4. $\displaystyle \frac{d}{dx}\,3x^{2}$$\displaystyle = 6x$
2. 02What is f''(x)?$\displaystyle f(x)=x^3-2x^2$AnswerHide

   Answer$\displaystyle 6x-4$

   Steps

   1. Differentiate once
   2. $\displaystyle \frac{d}{dx}\,(x^{3}-2x^2)$$\displaystyle = 3x^{2}-4x$
   3. Differentiate the result again
   4. $\displaystyle \frac{d}{dx}\,(3x^{2}-4x)$$\displaystyle = 6x-4$
3. 03What is f''(x)?$\displaystyle f(x)=x^4$AnswerHide

   Answer$\displaystyle 12x^2$

   Steps

   1. Differentiate once
   2. $\displaystyle \frac{d}{dx}\,x^{4}$$\displaystyle = 4x^{3}$
   3. Differentiate the result again
   4. $\displaystyle \frac{d}{dx}\,4x^{3}$$\displaystyle = 12x^{2}$
4. 04What is f''(x)?$\displaystyle f(x)=x^3+3x^2$AnswerHide

   Answer$\displaystyle 6x+6$

   Steps

   1. Differentiate once
   2. $\displaystyle \frac{d}{dx}\,(x^{3}+3x^2)$$\displaystyle = 3x^{2}+6x$
   3. Differentiate the result again
   4. $\displaystyle \frac{d}{dx}\,(3x^{2}+6x)$$\displaystyle = 6x+6$
5. 05What is f''(x)?$\displaystyle f(x)=x^3-3x^2$AnswerHide

   Answer$\displaystyle 6x-6$

   Steps

   1. Differentiate once
   2. $\displaystyle \frac{d}{dx}\,(x^{3}-3x^2)$$\displaystyle = 3x^{2}-6x$
   3. Differentiate the result again
   4. $\displaystyle \frac{d}{dx}\,(3x^{2}-6x)$$\displaystyle = 6x-6$
6. 06What is f''(x)?$\displaystyle f(x)=x^3+2x^2$AnswerHide

   Answer$\displaystyle 6x+4$

   Steps

   1. Differentiate once
   2. $\displaystyle \frac{d}{dx}\,(x^{3}+2x^2)$$\displaystyle = 3x^{2}+4x$
   3. Differentiate the result again
   4. $\displaystyle \frac{d}{dx}\,(3x^{2}+4x)$$\displaystyle = 6x+4$

FAQ

Q1What do you get when you differentiate $x^n$ twice?AnswerHide

A

Apply $(x^n)'=nx^{n-1}$ twice: $(x^n)''=n(n-1)x^{n-2}$. For $x^4$ that is $4\cdot3x^2=12x^2$; $x^2$ gives the constant $2$, and terms of degree 1 or less become $0$.