Second derivative

The second derivative of a function or polynomial is simply the derivative of its first derivative.

In other words, we differentiate twice!

It’s also sometimes said to be ‘the gradient of the gradient function’.

Basically, it tells us how the gradient of a function is changing as we move along the curve.

Notation

As mentioned in derivative notation, the second derivative of a function f(x) can be written as:

f''(x)

If we just have a polynomial y (in terms of x) then the second derivative is written as:

\frac{d^2y}{dx^2}

Notice the position of the squares!

Finding the second derivative

To find the second derivative we just differentiate it… and then differentiate it again!

It’s the derivative of the derivative.

Find the second derivative of f(x)=x^3+2x^2+x

• First, we find the first derivative:
  • f'(x)=3x^{3-1} + 2\times2x^{2-1} + 1x^{1-1}
  • =3x^2 + 4x + 1
• Now we differentiate it again to find the second derivative:
  • f''(x)=2\times3x^{2-1} + 1\times4x^{1-1} + 0
  • =6x + 4
• Answer: f''(x)=6x + 4

f'(x)=4x^3 - 2x + 7. Find f''(x).

• We differentiate f'(x) (the first derivative) to find the second derivative:
  • f''(x)=3\times4x^{3-1} - 1\times2x^{1-1} + 0
  • =12x^2 - 2
• Answer: f''(x)=12x^2 - 2

Find the second derivative of y=5x^4 - 3x^2 + x - 8.

• First, we find the first derivative:
  • \frac{dy}{dx}=4\times5x^{4-1} - 2\times3x^{2-1} + 1 - 0
  • =20x^3 - 6x + 1
• Now we differentiate it again to find the second derivative:
  • \frac{d^2y}{dx^2}=3\times20x^{3-1} - 1\times6x^{1-1} + 0
  • =60x^2 - 6
• Answer: \frac{d^2y}{dx^2}=60x^2 - 6

\frac{dy}{dx}=7x^5 + 4x^2 - x + 3. Find \frac{d^2y}{dx^2}.

• We differentiate \frac{dy}{dx} (the first derivative) to find the second derivative:
  • \frac{d^2y}{dx^2}=5\times7x^{5-1} + 2\times4x^{2-1} - 1 + 0
  • =35x^4 + 8x - 1
• Answer: \frac{d^2y}{dx^2}=35x^4 + 8x - 1