Calculus- Early Transcendentals, 2021a

4.4. Derivative Rules for Trigonometric Functions 135
Exercise 4.3.13 Compute the derivative of
Exercise 4.3.14 Compute the derivative of
x 2 + 5x − 3
x 5 − 6x 3 + 3x 2 − 7x + 1 .
x
√
x − 625
.
Exercise 4.3.15 Compute the derivative of
√
x − 5
x 20 .
Exercise 4.3.16 Find an equation for the tangent line to f (x)=(x 2 − 4)/(5 − x) at x = 3.
Exercise 4.3.17 Find an equation for the tangent line to f (x)=(x − 2)/(x 3 + 4x − 1) at x = 1.
Exercise 4.3.18 If f ′ (4)=5,g ′ (4)=12, ( fg)(4)= f (4)g(4)=2, and g(4)=6, compute f (4) and d dx
at 4.
f
g
4.4 Derivative Rules for Trigonometric Functions
We next look at the derivative of the sine function. In order to prove the derivative formula for sine, we
recall two limit computations from earlier:
sinx
cosx − 1
lim = 1 and lim = 0,
x→0 x
x→0 x
and the double angle formula
sin(A + B)=sinAcosB + sinBcosA.
Theorem 4.28: Derivative of Sine Function
(sinx) ′ = cosx
Proof. Let f (x)=sinx. Using the definition of derivative we have:
f ′ f (x + h) − f (x)
(x) = lim
h→0 h
= lim
h→0
sin(x + h) − sinx
h
= lim
h→0
sinxcosh + cosxsinh − sinx
h
cosh − 1
sinh
= lim sinx · lim + lim cosx · lim
h→0 h→0 h h→0 h→0 h
= sinx · 0 + cosx · 1
= cosx