MAT 21A Lecture Notes - Lecture 12: Product Rule, Quotient Rule
MAT 21A – Lecture 12 – Derivatives of Trigonometric Functions and Chain Rule
• Problem: What is derivative of the sin(x) function? That is, what is
?
=
. At this point it cannot be simplified nor algebraic
manipulation can be used. However, we can apply trig identities. In this case, we
can apply sum formula of sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
=
=
=
. Now we
need to rewrite
where we can multiply by its conjugate.
*
=
=
=
*
*
= [sin(x)](1*0) + cos(x) (1) = cos(x)
• Example: Find
and
= 4cos(x) + 2
=
• Problem: What is derivative of the cos(x) function? That is, what is
?
=
. Similarly, we can still use a trig identity which is
sum formula for cos(A + B) = cos(A)cos(B) – sin(A)sin(B)
=
=
=
=
= [cos(x)]*0 – sin(x)(1) = -sin(x)
• Also, you could think graphically about the derivatives of sin(x) and cos(x)
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