What a second derivative means
The second derivative is the derivative of the derivative. If you first find (f'(x)), then differentiating again gives (f''(x)). It measures how the slope is changing.
Method
- Differentiate once carefully. Rewrite the expression if needed so each term is easy to handle.
- Differentiate a second time. Apply the same rules again to the result.
- Simplify fully. Combine like terms, reduce fractions, and write the answer in its simplest form.
Important rules to use
- For powers of (x), lower the exponent by 1 each time you differentiate.
- Constants disappear when differentiated.
- If there are sums or differences, differentiate term by term.
- If the expression is more complicated, first simplify it before taking derivatives when possible.
Check your work
A good check is to differentiate your final (f''(x)) once and see whether you recover (f'(x)). Also, make sure signs and exponents change correctly at each step. The final answer should be exact and simplified.