Find out the answer to "Find the nth order derivative of the function..."

Tammy Todd

Answered question

2021-06-05

Find the nth order derivative of the function

y = 3^{2 x + 1}

dieseisB

Skilled2021-06-06Added 85 answers

Step 1
We use the chain rule to find the first 4 derivatives.

y = 3^{2 x + 1}

y^{(1)} = 3^{2 x + 1} (\ln 3) (2) = 2 \ln 3 \cdot 3^{2 x + 1}

y^{(2)} = 2 \ln 3 \cdot 3^{2 x + 1} (\ln 3) (2) = {(2 \ln 3)}^{2} \cdot 3^{2 x + 1}

y^{(3)} = {(2 \ln 3)}^{2} \cdot 3^{2 x + 1} (\ln 3) (2) = {(2 \ln 3)}^{3} \cdot 3^{2 x + 1}

y^{(4)} = {(2 \ln 3)}^{3} \cdot 3^{2 x + 1} (\ln 3) (2) = {(2 \ln 3)}^{4} \cdot 3^{2 x + 1}

Step 2
Using this pattern we get:

y^{(n)} = {(2 \ln 3)}^{n - 1} \cdot 3^{2 x + 1} (\ln 3) (2) = {(2 \ln 3)}^{n} \cdot 3^{2 x + 1}

Answer:

y^{(n)} = {(2 \ln 3)}^{n} \cdot 3^{2 x + 1}

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