Exercise
Given the following function:
f(x)=\ln x
Compute its nth derivative.
Final Answer
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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