# Maclaurin Series Derivatives – Useful Tables

 Derivative Sine Cosine $f^{ \left( 0 \right) }\left( x \right)$ $\sin { x }$ $\cos { x }$ $f^{ \left( 1 \right) }\left( x \right)$ $\cos { x }$ $-\sin { x }$ $f^{ \left( 2 \right) }\left( x \right)$ $-\sin { x }$ $-\cos { x }$ $f^{ \left( 3 \right) }\left( x \right)$ $-\cos { x }$ $\sin { x }$ $f^{ \left( 4 \right) }\left( x \right)$ $\sin { x }$ $\cos { x }$ $f^{ \left( 5 \right) }\left( x \right)$ $\cos { x }$ $-\sin { x }$ $f^{ \left( 6 \right) }\left( x \right)$ $-\sin { x }$ $-\cos { x }$ $f^{ \left( 7 \right) }\left( x \right)$ $-\cos { x }$ $\sin { x }$ $f^{ \left( 8 \right) }\left( x \right)$ $\sin { x }$ $\cos { x }$ $f^{ \left( 9 \right) }\left( x \right)$ $\cos { x }$ $-\sin { x }$
 Derivative Sine Cosine $f^{ \left( 0 \right) }\left( 0 \right)$ $0.0$ $1.0$ $f^{ \left( 1 \right) }\left( 0 \right)$ $1.0$ $0.0$ $f^{ \left( 2 \right) }\left( 0 \right)$ $0.0$ $-1.0$ $f^{ \left( 3 \right) }\left( 0 \right)$ $-1.0$ $0.0$ $f^{ \left( 4 \right) }\left( 0 \right)$ $0.0$ $1.0$ $f^{ \left( 5 \right) }\left( 0 \right)$ $1.0$ $0.0$ $f^{ \left( 6 \right) }\left( 0 \right)$ $0.0$ $-1.0$ $f^{ \left( 7 \right) }\left( 0 \right)$ $-1.0$ $0.0$ $f^{ \left( 8 \right) }\left( 0 \right)$ $0.0$ $1.0$ $f^{ \left( 9 \right) }\left( 0 \right)$ $1.0$ $0.0$

MACLAURIN SERIES EXPANSION:

SIN(X) FORMULA:

$\sin { x } ={ x }-\frac { { x }^{ 3 } }{ 3! } +\frac { { x }^{ 5 } }{ 5! } -\frac { { x }^{ 7 } }{ 7! } +\frac { { x }^{ 9 } }{ 9! } -...$

COS(X) FORMULA:

$\cos { x } =1-\frac { { x }^{ 2 } }{ 2! } +\frac { { x }^{ 4 } }{ 4! } -\frac { { x }^{ 6 } }{ 6! } +\frac { { x }^{ 8 } }{ 8! } -...$