Derivative of Cosine Function/Proof 4

Theorem

$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$

Proof

\(\ds \map {\frac \d {\d x} } {\cos x}\)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {\map \cos {x + h} - \cos x} h\)

Definition of Derivative of Real Function at Point

\(\ds \)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {\map \cos {\paren {x + \frac h 2} + \frac h 2} - \map \cos {\paren {x + \frac h 2} - \frac h 2} } h\)

\(\ds \)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {-2 \map \sin {x + \frac h 2} \map \sin {\frac h 2} } h\)

Simpson's Formula for Sine by Sine

\(\ds \)

\(=\)

\(\ds -\lim_{h \mathop \to 0} \map \sin {x + \frac h 2} \lim_{h \mathop \to 0} \frac {\map \sin {\frac h 2} } {\frac h 2}\)

Multiple Rule for Limits of Real Functions and Product Rule for Limits of Real Functions

\(\ds \)

\(=\)

\(\ds -\sin x \times 1\)

Real Sine Function is Continuous and Limit of $\dfrac {\sin x} x$ at Zero

\(\ds \)

\(=\)

\(\ds -\sin x\)

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Standard Differential Coefficients