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, start superscript, 3, end superscript, plus, 2, x, start superscript, 2, end superscript. Its first derivative is f, prime, left parenthesis, x, right parenthesis, equals, 3, x, start superscript, 2, end superscript, 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.
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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, start superscript, 2, end superscript, y, divided by, d, x, start superscript, 2, end superscript, end fraction. For example, the Leibniz notation for the second derivative of x, start superscript, 3, end superscript, plus, 2, x, start superscript, 2, end superscript is start fraction, d, start superscript, 2, end superscript, divided by, d, x, start superscript, 2, end superscript, end fraction, left parenthesis, x, start superscript, 3, end superscript, plus, 2, x, start superscript, 2, end superscript, right parenthesis.

Check your understanding

Problem 1
f, left parenthesis, x, right parenthesis, equals, 2, cosine, left parenthesis, start fraction, x, divided by, 2, end fraction, right parenthesis
f, start superscript, prime, prime, end superscript, left parenthesis, x, right parenthesis, equals, question mark
Choose 1 answer:
Choose 1 answer:

Problem 2
start fraction, d, start superscript, 2, end superscript, divided by, d, x, start superscript, 2, end superscript, end fraction, open bracket, start fraction, 10, divided by, 3, x, start superscript, 3, end superscript, end fraction, close bracket, equals, question mark
Choose 1 answer:
Choose 1 answer:

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