Question Type 3: Higher-order derivatives of functions with repeating patterns Exercises

Find $f^{(4)}(x)$ for $f(x)=e^{3x}$.

Evaluate $h^{(5)}\!\left(\tfrac{\pi}{4}\right)$ for $h(x)=\sin x-\cos x$. [4 marks]

Compute $y^{(5)}(x)$ for $y=\sin(2x)$.

Find the sixth derivative of $y=\cos(5x)$.

Evaluate $f^{(6)}(0)$ for $f(x)=\sin(3x)$.

Find the smallest positive integer $n$ such that $f^{(n)}(x)=f(x)$ for $f(x)=\sin x+\cos x$.

Evaluate $f^{(5)}(0)$ for $f(x)=\cos(2x)+\sin(3x)$.

Find $y^{(6)}(x)$ for $y=\sin x+2\sin\left(x+\frac{\pi}{3}\right)$.