Exercise
Given the following function:
f(x)=a^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)=a^x\cdot \ln a
We want to compute the second derivative. To do this, we derive the first derivative and get:
f''(x)=\ln a\cdot a^x\cdot \ln a=
=a^x\cdot {(\ln a)}^2=
Now, we want to compute the third derivative. To do this, again, we derive the second derivative and get:
f'''(x)={(\ln a)}^2 \cdot a^x\cdot \ln a=
=a^x\cdot {(\ln a)}^3=
Now one can see the pattern of the derivatives of the function, so the n-derivative is
f^{(n)}(x)=a^x\cdot {(\ln a)}^n
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