Calculus- Early Transcendentals, 2021a

154 Derivatives
Exercise 4.7.8 Find an equation for the tangent line to x 2/3 + y 2/3 = a 2/3 at a point (x 1 ,y 1 ) on the curve,
with x 1 ≠ 0 and y 1 ≠ 0. (Thiscurveisanastroid.)
Exercise 4.7.9 Find an equation for the tangent line to (x 2 +y 2 ) 2 = x 2 −y 2 at a point (x 1 ,y 1 ) on the curve,
with x 1 ≠ 0,−1,1. (Thiscurveisalemniscate.)
Exercise 4.7.10 Two curves are orthogonal if at each point of intersection, the angle between their
tangent lines is π/2. Two families of curves, A and B,areorthogonal trajectories of each other if given
any curve C in A and any curve D in B the curves C and D are orthogonal. For example, the family of
horizontal lines in the plane is orthogonal to the family of vertical lines in the plane.
(a) Show that x 2 −y 2 = 5 is orthogonal to 4x 2 +9y 2 = 72. (Hint: You need to find the intersection points
of the two curves and then show that the product of the derivatives at each intersection point is −1.)
(b) Show that x 2 + y 2 = r 2 is orthogonal to y = mx. Conclude that the family of circles centered at the
origin is an orthogonal trajectory of the family of lines that pass through the origin.
Note that there is a technical issue when m = 0. The circles fail to be differentiable when they cross
the x-axis. However, the circles are orthogonal to the x-axis. Explain why. Likewise, the vertical
line through the origin requires a separate argument.
(c) For k ≠ 0 and c ≠ 0 show that y 2 − x 2 = k is orthogonal to yx = c. In the case where k and c are
both zero, the curves intersect at the origin. Are the curves y 2 − x 2 = 0 and yx = 0 orthogonal to
each other?
(d) Suppose that m ≠ 0. Show that the family of curves {y = mx + b | b ∈ R} is orthogonal to the family
of curves {y = −(x/m)+c | c ∈ R}.
Exercise 4.7.11 Differentiate the function y = (x − 1)8 (x − 23) 1/2
27x 6 (4x − 6) 8
Exercise 4.7.12 Differentiate the function f (x)=(x + 1) sinx .
Exercise 4.7.13 Differentiate the function g(x)= ex (cosx + 2) 3
√ .
x 2 + 4
4.8 Derivatives of Inverse Functions
Suppose we wanted to find the derivative of the inverse, but do not have an actual formula for the inverse
function? Then we can use the following derivative formula for the inverse evaluated at a.
Derivative of f −1 (a)
Given an invertible function f (x), the derivative of its inverse function f −1 (x) evaluated at x = a is:
[
f
−1 ] ′
(a)=
1
f ′ [ f −1 (a)]