Learn what the second derivative tells us about how gradients change, concavity, and acceleration. Master this powerful tool for understanding the behavior of functions.
You already know that the first derivative of a function tells us the gradient at any point. But what if we want to know how that gradient is changing?
That's where the second derivative comes in!
The second derivative of a function is the derivative of the first derivative. It shows us the rate of change of the gradient.
In words: "Differentiate the first derivative"
Think About It:
If the first derivative tells us the velocity of a moving object, then the second derivative tells us the acceleration — how quickly the velocity is changing!
Just like the first derivative has multiple notations, the second derivative does too. Here are the main ways to write it:
If we have a function $f(x)$:
If we have $y = f(x)$:
Important Note:
$\frac{d^2y}{dx^2}$ does NOT mean $\frac{d^2y}{(dx)^2}$. It's a special notation that means "the second derivative of y with respect to x".
The second derivative gives us important information about how the function's gradient is changing:
Gradient is increasing
The curve is concave up (shaped like a smile or cup)
Gradient is constant
Possible point of inflection (where concavity changes)
Gradient is decreasing
The curve is concave down (shaped like a frown or cap)
Position: $s(t)$
Where the object is at time t
Velocity: $v(t) = s'(t)$
How fast the position is changing
Acceleration: $a(t) = v'(t) = s''(t)$
How fast the velocity is changing
At x = 0.0:
y = 0.00
First Derivative:
f'(0.0) = -1.50
Gradient = -1.50
Second Derivative:
f''(0.0) = 0.00
Inflection Point
• f''(x) > 0
• Gradient is increasing
• Tangent lines get steeper from left to right
• Curve opens upward like a cup
Think: water would collect in it
• f''(x) < 0
• Gradient is decreasing
• Tangent lines get less steep from left to right
• Curve opens downward like a cap
Think: water would run off it
Key Observations:
Finding the second derivative is straightforward: differentiate twice!
1Find the first derivative
Use the differentiation rules to find $f'(x)$ or $\frac{dy}{dx}$
2Differentiate again
Differentiate the first derivative to get $f''(x)$ or $\frac{d^2y}{dx^2}$
Key Insight:
The degree of the polynomial decreases by 1 each time you differentiate. For example:
• $x^3$ → $3x^2$ → $6x$
• Cubic → Quadratic → Linear
Step 1: Find the first derivative
Step 2: Find the second derivative
Answer:
$k''(x) = 12x - 8$
Step 1: Rewrite using negative exponents
Step 2: Find the first derivative
Step 3: Find the second derivative
Answer:
$\frac{d^2y}{dx^2} = \frac{6}{x^3}$
Step 1: Expand the function
Step 2: Find the first derivative
Step 3: Find the second derivative
Answers:
$f'(x) = 2x - 6$
$f''(x) = 2$
Notice that the second derivative is a constant! This makes sense because the original function is a quadratic, so its gradient changes at a constant rate.
Original Function f(x)
Shows the shape of the curve
First Derivative f'(x)
Shows the gradient at each point
Second Derivative f''(x)
Shows rate of change of gradient
Observe:
Apply what you've learned with these practice problems.
Test your understanding with these questions.
"The second derivative tells us how the rate of change is changing."
Key Formulas:
$f''(x) = \frac{d}{dx}[f'(x)]$
$\frac{d^2y}{dx^2} = \frac{d}{dx}\left[\frac{dy}{dx}\right]$
What It Tells Us:
Remember:
Differentiate twice, using the same rules each time. The degree of the polynomial decreases with each derivative.