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Calculating Derivative – Computing nth derivative – Exercise 1079

Exercise

Given the following function:

f(x)=a^x

Compute its nth derivative.

Final Answer


f^{(n)}(x)=a^x\cdot {(\ln a)}^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)=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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