Calculus- Early Transcendentals, 2021a

134 Derivatives
Exercises for Section 4.3
Exercise 4.3.1 Find the derivatives of the following functions.
(f) x −9/7
(a) x 100
(e) x 3/4 (j) (x + 1)(x 2 + 2x − 3)
(b) x −100
(g) 5x 3 + 12x 2 − 15
(c)
1
x 5
(h) −4x 5 + 3x 2 − 5/x 2
(d) x π
(i) 5(−3x 2 + 5x + 1)
(k) (x + 1)(x 2 + 2x − 3) −1
(l) x 3 (x 3 − 5x + 10)
(m) (x 2 + 5x − 3)(x 5 )
(n) (x 2 + 5x − 3)(x −5 )
(o) (5x 3 + 12x 2 − 15) −1
Exercise 4.3.2 Find an equation for the tangent line to f (x)=x 3 /4 − 1/xatx= −2.
Exercise 4.3.3 Find an equation for the tangent line to f (x)=3x 2 − π 3 at x = 4.
Exercise 4.3.4 Suppose the position of an object at time t is given by f (t)=−49t 2 /10 + 5t + 10. Finda
function giving the speed of the object at time t. The acceleration of an object is the rate at which its speed
is changing, which means it is given by the derivative of the speed function. Find the acceleration of the
object at time t.
Exercise 4.3.5 Let f (x)=x 3 and c = 3. Sketch the graphs of f , c f , f ′ , and (cf) ′ on the same diagram.
Exercise 4.3.6 The general polynomial P of degree n in the variable x has the form P(x) =
a 0 + a 1 x + ...+ a n x n . What is the derivative (with respect to x) of P?
Exercise 4.3.7 Find a cubic polynomial whose graph has horizontal tangents at (−2,5) and (2,3).
Exercise 4.3.8 Prove that
d
dx (cf(x)) = cf′ (x) using the definition of the derivative.
n
∑ a k x k =
k=0
Exercise 4.3.9 Suppose that f and g are differentiable at x. Show that f − g is differentiable at x using
the two linearity properties from this section.
Exercise 4.3.10 Use the product rule to compute the derivative of f (x)=(2x − 3) 2 . Sketch the function.
Find an equation of the tangent line to the curve at x = 2. Sketch the tangent line at x = 2.
Exercise 4.3.11 Suppose that f , g, and h are differentiable functions. Show that ( fgh) ′ (x)= f ′ (x)g(x)h(x)+
f (x)g ′ (x)h(x)+ f (x)g(x)h ′ (x).
Exercise 4.3.12 Compute the derivative of
x 3
x 3 − 5x + 10 .