Second derivative

Key takeaways

Second derivative belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Second derivative to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Second derivative from memory before moving on to harder problems.

Reference excerpt

In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation:

a = d v d t = d 2 x d t 2 , {\displaystyle a={\frac {dv}{dt}}={\frac {d^{2}x}{dt^{2}}},}

where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change. The last expression d 2 x d t 2 {\displaystyle {\tfrac {d^{2}x}{dt^{2}}}} is the second derivative of position (x) with respect to time. On the graph of a function, the sign of the second derivative is related to the concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way.

Second derivative power rule The power rule for the first derivative, if applied twice, will produce the second derivative power rule as follows:

d 2 d x 2 x n = d d x d d x x n = d d x ( n x n − 1 ) = n d d x x n − 1 = n ( n − 1 ) x n − 2 . {\displaystyle {\frac {d^{2}}{dx^{2}}}x^{n}={\frac {d}{dx}}{\frac {d}{dx}}x^{n}={\frac {d}{dx}}\left(nx^{n-1}\right)=n{\frac {d}{dx}}x^{n-1}=n(n-1)x^{n-2}.}

Notation

The second derivative of a function f ( x ) {\displaystyle f(x)} is usually denoted f ″ ( x ) {\displaystyle f''(x)} . That is:

f ″ = ( f ′ ) ′ {\displaystyle f''=\left(f'\right)'}

When using Leibniz's notation for derivatives, the second derivative of a dependent variable y with respect to an independent variable x is written

d 2 y d x 2 . {\displaystyle {\frac {d^{2}y}{dx^{2}}}.}

This notation is derived from the following formula:

d 2 y d x 2 = d d x ( d y d x ) . {\displaystyle {\frac {d^{2}y}{dx^{2}}}\,=\,{\frac {d}{dx}}\left({\frac {dy}{dx}}\right).}

Example Given the function

f ( x ) = x 3 , {\displaystyle f(x)=x^{3},}

the derivative of f is the function

f ′ ( x ) = 3 x 2 . {\displaystyle f'(x)=3x^{2}.}

The second derivative of f is the derivative of f ′ {\displaystyle f'} , namely

… excerpt ends here. Continue reading the full article.

Illustrations

Second derivative: A plot of

f
(
x
)
=
sin
⁡
(
2
x
)

{\displaystyle f(x)=\sin(2x)}

from

−
π

/

4

{\displaystyle -\pi /4}

to

5
π

/

4

{\displaystyle 5\pi /4}

. The tangent line is blue where the curve is concave up, green where the curve is concave down, and red at the inflection points (0,

π

{\displaystyle \pi }

/2, and

π

{\displaystyle \pi }

). — A plot of f ( x ) = sin ⁡ ( 2 x ) {\displaystyle f(x)=\sin(2x)} from − π / 4 {\displaystyle -\pi /4} to 5 π / 4 {\displaystyle 5\pi /4} . The tangent line is blue where the curve is concave up, green where the curve is concave down, and red at the inflection points (0, π {\displaystyle \pi } /2, and π {\displaystyle \pi } ).

Worked examples

Example 1 — a first encounter with Second derivative

Start with the simplest possible case. Write down what Second derivative claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Second derivative before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Second derivative ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Second derivative

In research: Second derivative appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Second derivative in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Second derivative is common in secondary-school and first-year university syllabi. It links to neighbouring topics Differential calculus, Functions and mappings, Linear operators in calculus, so understanding it makes those chapters shorter.
In everyday life: Look for Second derivative outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Second derivative in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Second derivative means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Second derivative out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Second derivative in simple terms?

In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect…

Why does Second derivative matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Second derivative?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Second derivative.