Higher Derivatives

Reading

• Sections 3.5

Practice problems

• Section 3.5: 3, 9, 13, 23
• To turn in (together with 3.3): 3.5 12, 14

Notes

Higher Derivatives

Since the first derivative is a function itself, we can take its derivative. We call this the second derivative:

Second Derivative \[\frac{d^2 f}{dx^2} = f''(x) = \frac{d}{dx}\left(f'(x)\right)\]

Example: Let \(f(x) = 3x^3-x^2+1\). Then \(f'(x) = 9x^2-2x\). Then the derivative of that is \(f''(x) = 18x-2\).

We can similarly take higher-order derivatives:

Higher Order Derivatives \[f^{(3)}(x) = \frac{d}{dx}\left(f''(x)\right)\] \[f^{(4)}(x) = \frac{d}{dx}\left(f^{(3)}(x)\right)\] and so on.

Practice

• Calculate the first four derivatives of \(f(x) = \frac{1}{x}\).
• Calculate the first four derivatives of \(2x^3-2x^2+x-1\).
• Calculate the first four derivatives of \(\sin(x)\).