Calculus- Early Transcendentals, 2021a

158 Derivatives
Exercise 4.8.4 The inverse cosecant function, denoted by csc −1 (x), is defined to be the inverse of the
restricted cosecant function: csc(x), x∈ (0,π/2] ∪ (π,3π/2]. Find the derivative of csc −1 (x).
Exercise 4.8.5 Suppose f (x) =x 3 + 4x + 2. Find the slope of the tangent line to the graph of g(x) =
xf −1 (x) at the point where x = 7.
Exercise 4.8.6 Find the derivatives of sin −1 (x)+cos −1 (x) and (x 2 + 1)tan −1 (x).
Exercise 4.8.7 Differentiate y = sin −1 (x 2 ) and y = tan −1 (3x).
4.9 Additional Exercises
Exercise 4.9.1 Find the derivatives of the following functions from definition.
(a) f (x)=(2x + 3) 2
(b) g(x)=x 3/2
Exercise 4.9.2 Let f (x)=
Exercise 4.9.3 Differentiate the following functions.
(a) y = 7x 4 − 7π 4 + 1
π 3√ x
(b) f (x)= 1 − √ x
1 + √ x
(c) f (x)=|x − 1| + |x + 2|
(d) f (x)=x 2 sinxcosx
(e) y =
xsinx
1 + sinx
√
(f) g(x)= 2 + √ 3 x
(g) y = 3√ x 4 + x 2 + 1 +
(h) y = sin 3 x − sin(x 3 )
(i) F(x)=sec 4 x + tan 4 x
( ) 1 − x
(j) y = cos 2 1 + x
{ x
3
if x ≤ 1
5x − x 2 if x > 1 . Use the definition of the derivative to find f ′ (1).
1
(x 3 − x + 4) 5