Derivatives of Trig Functions

Suggested Prerquesites:

Useful Trigonometric Identities, The derivative of sine, The derivative of cosine, The Quotient Rule, The Chain Rule

Given that we know the derivatives of sine and cosine, we can readily compute the derivatives of the other trigonometric functions, since they can be defined in terms of sine and cosine.

d dx tan (x) &sp;=&sp; d dx (sin (x) cos (x)) &sp;=&sp; cos^2^(x)
 + sin^2^(x) cos^2^(x) &sp;=&sp; 1 cos^2^(x) &sp;=&sp; sec^2^(x)

$d dx ctn (x) &sp;=&sp; d dx (cos (x) sin (x)) &sp;=&sp; - sin^2^(x)
- cos^2^(x) sin^2^(x) &sp;=&sp; -1 sin^2^(x) &sp;=&sp;
- csc^2^(x)$

$d dx sec (x) &sp;=&sp; d dx (cos (x) )^-1^ &sp;=&sp;
(-1)(cos (x) )^-2^(- sin (x) ) &sp;=&sp; tan (x) sec (x)$

$d dx csc (x) &sp;=&sp; d dx (sin (x) )^-1^ &sp;=&sp;
(-1)(sin (x) )^-2^cos (x) &sp;=&sp;
- ctn (x) csc (x)$

Some examples::

$d dx cos (x) sin (x) &sp;=&sp;
- sin^2^(x) + cos^2^(x)$
$d dt tan (4x^2^) &sp;=&sp;
8x&sp; sec^2^(4x^2^)$
$d ds sin (cos (s) )
&sp;=&sp; - sin (s) cos (cos (s) )$

Exercises:

$y &sp;=&sp; 4 csc (-6x)$
$s &sp;=&sp; sec^2^(r) - tan^2^(r)$
$w &sp;=&sp; tan^2^(sin (v) )$

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