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 $f(x)=\sqrt{x}$.