Calculating Derivative – Computing nth derivative – Exercise 1079

Exercise

Given the following function:

$f(x)=a^x$

Compute its nth derivative.

Final Answer

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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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