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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 3

Lesson 8: Calculating higher-order derivatives

# Second derivatives review

Review your knowledge of second derivatives.

## What are second derivatives?

The second derivative of a function is simply the derivative of the function's derivative.
Let's consider, for example, the function f, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2, x, squared. Its first derivative is f, prime, left parenthesis, x, right parenthesis, equals, 3, x, squared, plus, 4, x. To find its second derivative, f, start superscript, prime, prime, end superscript, we need to differentiate f, prime. When we do this, we find that f, start superscript, prime, prime, end superscript, left parenthesis, x, right parenthesis, equals, 6, x, plus, 4.

## Notation for second derivatives

We already saw Lagrange's notation for second derivative, f, start superscript, prime, prime, end superscript.
Leibniz's notation for second derivative is start fraction, d, squared, y, divided by, d, x, squared, end fraction. For example, the Leibniz notation for the second derivative of x, cubed, plus, 2, x, squared is start fraction, d, squared, divided by, d, x, squared, end fraction, left parenthesis, x, cubed, plus, 2, x, squared, right parenthesis.