 # Calculating Derivative – Computing nth derivative – Exercise 1084

Exercise

Given the following function:

$$f(x)=\ln x$$

Compute its nth derivative.

$$f^{(n)}(x)={(-1)}^{n-1}(n-1)! x^{-n}$$

Solution

We compute the first derivatives and try to find a pattern for the nth derivative.

Using Derivative formulas, we get the first derivative:

$$f'(x)=\frac{1}{x}$$

We want to compute the second derivative. To do this, we derive the first derivative and get:

$$f''(x)=-\frac{1}{x^2}=-x^{-2}$$

Now, we want to compute the third derivative. To do this, again, we derive the second derivative and get:

$$f'''(x)=2x^{-3}$$

Next, we want to compute the forth derivative. To do this, again, we derive the third derivative and get:

$$f^{(4)}(x)=-6x^{-4}$$

Now we look at the derivatives of the function:

$$f'(x)=\frac{1}{x}=x^{-1}$$

$$f''(x)=-\frac{1}{x^2}=(-1)\cdot x^{-2}$$

$$f'''(x)=2x^{-3}=1\cdot 2 x^{-3}$$

$$f^{(4)}(x)=(-1)\cdot 1\cdot 2\cdot 3 x^{-4}$$

One can see the pattern of the derivatives of the function, so the n-derivative is

$$f^{(n)}(x)={(-1)}^{n-1}(n-1)! x^{-n}$$

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