Applications of Derivatives

The Second Derivative

Find f''(x)=6x by differentiating f'(x)=3x²-3 for f(x)=x³-3x.

Example

Differentiate the first derivative to get the second derivative. The first derivative reports how fast the quantity is changing; differentiating again measures how fast that rate itself changes. This second derivative is what tells you whether a graph curves upward or downward and how quickly the slope is turning, which is why it captures acceleration and concavity.

highlighted = computed this step

Step 1 — Set up

Start from the first derivative.

f′(x)=3x2−3f'(x)= 3x^{2}-3

Step 2 — Differentiate again

Differentiate f prime again.

ddx3x2−3\frac{d}{dx} \hlmath{3x^{2}-3}

Step 3 — Second derivative

State the second derivative.

f′′(x)=6xf''(x)= \hlmath{6x}
second-derivative The second derivative f''(x) is the derivative of f'(x). It measures the rate of change of the slope and is used to analyze concavity and to classify critical points via the second derivative test.