Question

question
find $\frac{d^{15}}{dx^{15}}(- 5sin(x)+13cos(x))$.
provide your answer below:
$\frac{d^{15}}{dx^{15}}(-5sin(x)+13cos(x))=square$

Explanation:

Step1: Recall derivative rules for sin and cos

The derivative of $\sin(x)$ is $\cos(x)$ and the derivative of $\cos(x)$ is $-\sin(x)$. The $n$-th derivative of $\sin(x)$ and $\cos(x)$ follow a cycle.

Step2: Determine the cycle of derivatives

The derivatives of $\sin(x)$:
$y = \sin(x)$, $y'=\cos(x)$, $y''=-\sin(x)$, $y''' = -\cos(x)$, $y^{(4)}=\sin(x)$. The cycle has a period of 4.
For $\cos(x)$: $y=\cos(x)$, $y'=-\sin(x)$, $y'' = -\cos(x)$, $y'''=\sin(x)$, $y^{(4)}=\cos(x)$.

Step3: Find the 15 - th derivative of $\sin(x)$

Divide 15 by 4: $15\div4 = 3\cdots\cdots3$. So the 15 - th derivative of $\sin(x)$ is the same as the 3 - rd derivative in the cycle, which is $-\cos(x)$.

Step4: Find the 15 - th derivative of $\cos(x)$

Divide 15 by 4: $15\div4=3\cdots\cdots3$. So the 15 - th derivative of $\cos(x)$ is the same as the 3 - rd derivative in the cycle, which is $\sin(x)$.

Step5: Calculate the 15 - th derivative of the given function

We have $y=- 5\sin(x)+13\cos(x)$. Using the linearity of differentiation $\frac{d^{15}}{dx^{15}}(-5\sin(x)+13\cos(x))=-5\frac{d^{15}}{dx^{15}}\sin(x)+13\frac{d^{15}}{dx^{15}}\cos(x)$.
Substitute the values from Step 3 and Step 4: $-5(-\cos(x)) + 13\sin(x)=5\cos(x)+13\sin(x)$.

Answer:

$5\cos(x)+13\sin(x)$