Find the "N" order derivative of the following functions.

Question

Identify the pattern and express a formula to calculate Yⁿ:

1.- f(x)= 1/(3x)²

2.- f(x)= log(5x)

Answer 1

Hi Greg: here's how to solve this one.

Do the first 3 or 4 derivatives and see the pattern of the constant (usually with an n!), the algebraic sign (usually (-1)ⁿ or (-1)^{n + 1}, and the power of x.

1) f(x) = 1/(3x)² = (1/9) x^-2 [Always change powers of x below the line, to negative powers of x; easier

=> f'(x) = (1/9) (-2) x^-3

=> f''(x) = (1/9) (-3) (-2) x^-4 = (1/9) (3 . 2) x^-4

=> f'''(x) = (1/9) (-4 . 3. 2) x^-5

Now, we see the pattern: the signs alternate with odd powers corresponding to negative and even powers to positive. For the n-th derivative, the constant is (n + 1)! and the power of x is -(n + 3)

Therefore, f⁽ⁿ⁾(x) = (-1)ⁿ( n + 1)! x^{-(n + 3)}

2) f(x) = log(5x) = log 5 + log x [by the laws of logs

=> f'(x) = 1/x = -1 x^-1 the derivative of log 5 is, of course, zero

=> f''(x) = 1 x^-2

=> f'''(x) = -2 x^-3

=> f^(iv)(x) = +(3 . 2) x^-4

And we see the pattern: f⁽ⁿ⁾(x) = (-1)ⁿ (n - 1)! x^-n