World Web Math: Derivative of Cosine

Suggested Prerquesites:

Definition of the Derivative, Introduction to Trigonometric Functions, Useful Trigonometric Identities, Evaluating Limits

We'll deduce the derivative of cos(x) directly from the definition of the derivitive.

d dx cos (x) &sp;=&sp; lim h→0 cos (x+h) - cos (x) h

$&sp;=&sp; lim h→0 cos (h) cos (x) - sin (h) sin (x) - cos (x) h$

$cos (x) (lim h→0 cos (h) -1 h) &sp;-&sp; sin (x) (lim h→0 sin (x) h)$

Then, since $lim h→0 cos (h) -1 h &sp;=&sp; 0$ and $lim h→0 sin (x) h &sp;=&sp; 1$

$d dx cos (x) &sp;=&sp;
- sin (x)$

Does this make sense? Cosine is increasing where sine is negative and decreasing where sine is positive. The tangent lines to cosine are flat where sine is zero. So, it does make sense.

Exercises:

$d dt 4 cos (t) &sp;=&sp; ?$
$d dr (r^2^cos (r)) &sp;=&sp; ?$
Prove that $D_x_sin (2x) &sp;=&sp;
2 cos (2x)$
(Hint: use double angle formulas)

Solutions to the exercises | Back to the Calculus page | Back to the World Web Math top page

jjnichol@mit.edu