The Second Derivative Revision Notes for VCE SSCE Mathematical Methods

The Second Derivative

Understanding the second derivative

The second derivative is simply the derivative of the derivative. When we differentiate a function once, we get the first derivative. When we differentiate that result again, we obtain the second derivative.

Let's consider a concrete example to understand this concept. Suppose we have the function $f(x) = 2x^2 + 4x + 1$ .

When we differentiate this function once, we find the first derivative:

f'(x) = 4x + 4

To find the second derivative, we differentiate the first derivative:

The derivative of $4x + 4$ is $4$

Therefore, the second derivative is $f''(x) = 4$ .

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Notice that the second derivative tells us how the rate of change itself is changing. It gives us information about the curvature or concavity of the original function.

Notation for second derivatives

There are two main ways to express the second derivative, and you should be comfortable with both notations.

Function notation

When using function notation, we write the second derivative as f''(x).

The double prime symbol $''$ indicates that we have differentiated twice.

For example: if $f(x) = 2x^2 + 4x + 1$ , then $f''(x) = 4$

Leibniz notation

When using Leibniz notation, we write the second derivative as $\frac{d^2y}{dx^2}$ .

This notation emphasises that we are taking the derivative twice with respect to $x$ .

For example: if $y = 2x^2 + 4x + 1$ , then:

First derivative: $\frac{dy}{dx} = 4x + 4$
Second derivative: $\frac{d^2y}{dx^2} = 4$

chatImportant

Both notations represent exactly the same mathematical operation, so use whichever is most appropriate for the context.

Worked example

Let's work through finding the second derivative for several different types of functions. This will help you develop confidence with the process.

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Worked Example: Finding Second Derivatives

Part a: Find $f''(x)$ for $f(x) = 3x^3 + 2x^{-1} + 1$ , where $x \neq 0$

First, we find the first derivative using the power rule:

f'(x) = 9x^2 - 2x^{-2} \text{ for } x \neq 0

Now we differentiate again to find the second derivative:

f''(x) = 18x + 4x^{-3} \text{ for } x \neq 0

Part b: Find $f''(x)$ for $f(x) = x^{\frac{1}{2}} + 3x^{-4} + 1$ , where $x > 0$

First, we find the first derivative:

f'(x) = \frac{1}{2}x^{-\frac{1}{2}} - 12x^{-5} \text{ for } x > 0

Now we differentiate again:

f''(x) = -\frac{1}{4}x^{-\frac{3}{2}} + 60x^{-6} \text{ for } x > 0

Part c: Find $f''(x)$ for $f(x) = x^4 + x^{-\frac{3}{2}} + 1$ , where $x > 0$

First, we find the first derivative:

f'(x) = 4x^3 - \frac{3}{2}x^{-\frac{5}{2}} \text{ for } x > 0

Now we differentiate again:

f''(x) = 12x^2 + \frac{15}{4}x^{-\frac{7}{2}} \text{ for } x > 0

Notice in each case that we apply the power rule twice. Be careful with negative and fractional powers - the algebra can become tricky, so work methodically and check your calculations.

Application to position, velocity and acceleration

The second derivative has an important physical interpretation when we're analysing motion in a straight line.

Suppose an object is moving along a straight line, and we know its position at any time $t$ . We write this position function as $x(t)$ .

The first derivative $\frac{dx}{dt}$ gives us the velocity of the object at time $t$ . Velocity tells us how quickly the position is changing.

The second derivative $\frac{d^2x}{dt^2}$ gives us the acceleration of the object at time $t$ . Acceleration tells us how quickly the velocity is changing.

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This creates a useful chain of relationships:

Position function: $x(t)$
Velocity (first derivative): $\frac{dx}{dt}$
Acceleration (second derivative): $\frac{d^2x}{dt^2}$

Understanding this connection helps you see why the second derivative matters in physics and real-world applications. Whenever you need to analyse how things are speeding up or slowing down, you're working with second derivatives.

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Key Points to Remember:

The second derivative is found by differentiating twice - first find $f'(x)$ , then differentiate again to get $f''(x)$
Two notations are commonly used: function notation f''(x) and Leibniz notation $\frac{d^2y}{dx^2}$
The second derivative tells us about the rate of change of the rate of change
In motion problems, the second derivative of position gives acceleration
Apply the power rule carefully when dealing with negative and fractional powers

The Second Derivative (VCE SSCE Mathematical Methods): Revision Notes