Second Derivative Revision Notes for Grade 12 NSC Matric Mathematics

Second Derivative

What is the second derivative?

The second derivative is found by differentiating the first derivative of a function. It provides important information about how the gradient of the original function is changing.

Definition: The second derivative measures the rate of change of the gradient of a function. It tells us whether the gradient is increasing, decreasing, or staying constant.

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The second derivative gives us insight into the curvature and concavity of a function, making it essential for understanding the shape of graphs and optimising functions.

Mathematical notation

There are two main ways to write the second derivative:

Function notation: $f''(x)$ (read as "f double prime of x")
Leibniz notation: $\frac{d^2y}{dx^2}$ (read as "d squared y, d x squared")

The mathematical relationship is: $f''(x) = \frac{d}{dx}[f'(x)]$

This can also be written as: $y'' = \frac{d}{dx}\left[\frac{dy}{dx}\right] = \frac{d^2y}{dx^2}$

How to find the second derivative

The process of finding the second derivative involves applying differentiation rules twice in succession.

Step 1: Find the first derivative $f'(x)$ using standard differentiation rules Step 2: Differentiate the first derivative to get $f''(x)$

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Remember that each differentiation step follows the same rules you've learned - power rule, product rule, chain rule, etc. The key is to be systematic and careful with your algebra.

Worked examples

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Worked Example 1: Polynomial function

Question: Find the second derivative of $k(x) = 2x^3 - 4x^2 + 9$

Solution:

Step 1: Find the first derivative

$k'(x) = 2(3x^2) - 4(2x) + 0$

$k'(x) = 6x^2 - 8x$

Step 2: Differentiate again to find the second derivative

$k''(x) = 6(2x) - 8(1)$

$k''(x) = 12x - 8$

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Worked Example 2: Function with negative powers

Question: Find the second derivative of $y = \frac{3}{x}$

Solution:

Step 1: Rewrite using negative powers

$y = 3x^{-1}$

Step 2: Find the first derivative

$\frac{dy}{dx} = 3(-1x^{-2})$
$\frac{dy}{dx} = -3x^{-2}$
$\frac{dy}{dx} = -\frac{3}{x^2}$

Step 3: Find the second derivative

$\frac{d^2y}{dx^2} = -3(-2x^{-3})$
$\frac{d^2y}{dx^2} = 6x^{-3}$
$\frac{d^2y}{dx^2} = \frac{6}{x^3}$

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Worked Example 3: Quadratic function

Question: Find the second derivative of $f(x) = 5x^2 + 3x - 7$

Solution:

Step 1: Find the first derivative

$f'(x) = 5(2x) + 3(1) - 0$
$f'(x) = 10x + 3$

Step 2: Find the second derivative

$f''(x) = 10(1) + 0$
$f''(x) = 10$

Exam tips

Always differentiate twice: The second derivative requires two separate differentiation steps
Check your algebra: Simple arithmetic errors are common when working with multiple terms
Rewrite fractions: Convert expressions like $\frac{3}{x}$ to $3x^{-1}$ before differentiating
Constant rule: Remember that the derivative of a constant is zero

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Take your time with each step and show all working clearly. Examiners award marks for correct method even if the final answer contains a small error.

Common mistake to avoid

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Don't confuse the second derivative with squaring the first derivative. The second derivative is $f''(x) = \frac{d}{dx}[f'(x)]$ , not $[f'(x)]^2$ .

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Key Points to Remember:

The second derivative is the derivative of the first derivative
Use notation $f''(x)$ or $\frac{d^2y}{dx^2}$ to represent second derivatives
Differentiate twice: First find $f'(x)$ , then differentiate again to get $f''(x)$
The second derivative tells us about the rate of change of the gradient
Polynomial functions become simpler with each differentiation, whilst negative power functions become more complex

Second Derivative (Grade 12 NSC Matric Mathematics): Revision Notes