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Calculating Derivative – Computing nth derivative – Exercise 1084


Given the following function:

f(x)=\ln x

Compute its nth derivative.

Final Answer

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


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

Using Derivative formulas, we get the first derivative:


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


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


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


Now we look at the derivatives of the function:


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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