find the 80th derivative of f(x) = sin(x) by finding the first few derivatives and observing the pattern…
find the 80th derivative of f(x) = sin(x) by finding the first few derivatives and observing the pattern that occurs. f^(80)(x) =
Answer
Explanation:
Step1: Find first - few derivatives
$f(x)=\sin(x)$; $f'(x)=\cos(x)$; $f''(x)=-\sin(x)$; $f'''(x)=-\cos(x)$; $f^{(4)}(x)=\sin(x)$.
Step2: Identify the pattern
The derivatives of $\sin(x)$ have a cycle of 4. That is, $f^{(n + 4)}(x)=f^{(n)}(x)$ for all non - negative integers $n$.
Step3: Divide the order of the derivative by 4
We want to find the 80th derivative. Divide 80 by 4: $80\div4 = 20$ with a remainder of 0.
Answer:
$\sin(x)$