The Second Derivative

Introduction

The second derivative is simply the derivative of the derivative.
If a function f(x) represents a curve, then:

• The first derivative, $$f^\prime(x)$$, tells us the slope of the tangent line (rate of change).

• The second derivative, $$f^{\prime\prime}(x)$$, tells us how the slope itself changes — that is, the rate of change of the rate of change.

Definition

If

$$f^\prime(x)=\frac{dx}{dy}$$​

then the second derivative is written as:

$$f^{\prime\prime}(x) = \frac{d^{2}y}{dx^{2}}=\frac{d}{dx}\frac{dy}{dx}$$

It measures how $$f^\prime(x)$$ changes with respect to x.

Geometric Meaning

• $$f^\prime(x)$$ → Slope of the curve

• $$f^{\prime\prime}(x)$$ → Curvature of the curve

If:

• $$f^{\prime\prime}(x) > 0$$, the curve is concave up (looks like a “U”)

• $$f^{\prime\prime}(x) < 0$$, the curve is concave down (looks like an “∩”)

• $$f^{\prime\prime}(x) = 0$$, may indicate an inflection point, where the curve changes concavity.

Finding the Second Derivative

Let’s go step by step with examples:

Example 1

$$f^(x)=x^3-2x^2+4x$$

1️⃣ First derivative:

$$f^\prime(x) = 3x^2-4x$$

2️⃣ Second derivative:

$$f^{\prime\prime}(x) = 6x-4$$

Example 2

$$f(x)=\sin(x)$$

1️⃣ First derivative:

$$f^\prime(x) = \cos(x)$$

2️⃣ Second derivative:

$$f^{\prime\prime}(x) = -\sin(x)$$

Example 3

$$f(x) = e^{x}$$

1️⃣ First derivative:

$$f^\prime(x) = e^{x}$$

2️⃣ Second derivative:

$$f^{\prime\prime}(x) = e^{x}$$

So, the exponential function’s derivative and second derivative are the same as the original function.