# Math Insight

### The second and higher derivatives

The second derivative of a function is simply the derivative of the derivative. The third derivative of a function is the derivative of the second derivative. And so on.

The second derivative of a function $y=f(x)$ is written as $$y''=f''(x)={d^2\over dx\,^2}f={d^2\,f\over dx\,^2}={d^2\,y\over dx\,^2}$$ The third derivative is $$y''' =f'''(x)={d^3\over dx\,^3}f={d^3\,f\over dx\,^3}={d^3\,y\over dx\,^3}$$ And, generally, we can put on a ‘prime’ for each derivative taken. Or write $${d^n\over dx\,^n}f={d^n\,f\over dx\,^n}={d^n\,y\over dx\,^n}$$ for the $n$th derivative. There is yet another notation for high order derivatives where the number of ‘primes’ would become unwieldy: $${d^n\,f\over dx\,^n}=f^{(n)}(x)$$ as well.

The geometric interpretation of the higher derivatives is subtler than that of the first derivative, and we won't do much in this direction, except for the next little section.

#### Exercises

1. Find $f''(x)$ for $f(x)=x^3-5x+1$.
2. Find $f''(x)$ for $f(x)=x^5-5x^2+x-1$.
3. Find $f''(x)$ for $f(x)=\sqrt{x^2-x+1}$.
4. Find $f''(x)$ for $f(x)=\sqrt{x}$.