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Math 151, Benjamin
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3.5 The Chain Rule
To find the derivative of a function that is the composition of two functions for which we already know the
derivatives, we can use the Chain Rule.
The Chain Rule: Suppose F (x) = f (g(x)). Then, provided that g 0 (x) and f 0 (g(x)) both exist,
F 0 (x) = f 0 (g(x))g 0 (x)
Alternate way of thinking about it: If y = f (u) and u = g(x) where both are differentiable functions, then
dy
dy du
=
dx
du dx
Find the derivatives of the following functions.
• f (x) =
√
x2 + 9
2
• g(t) = 3t + 4
t
• f (x) =
• g(x) =
2
5x − 7
4x3 − x
q
4
5
5x4 +
(2t − cos t)7
10
√
6x + 1
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Math 151, Benjamin
Aurispa
• f (x) = tan 5x sec 3x + cos(x3 ) + sin2 x + cos4 2x
• f (x) = tan3 (csc 4x)
• h(x) = sec(sin x)
• Suppose that F (x) = f (g(x)) where f 0 (4) = 3, f 0 (5) = 2, g 0 (4) = 6 and g(4) = 5.
(a) Calculate F 0 (4).
(b) Calculate G0 (0) where G(x) = f (sin x + 5) + (g(5x + 4))2
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Math 151, Benjamin
Aurispa
3.6 Implicit Differentiation
Consider the equation y 3 + 2xy = 9. How would you calculate
dy
dx ?
When y cannot be written explicitly as a function of x (or not easily), we can use the method of implicit
differentiation.
dy
To find dx
, differentiate both sides with respect to x, remembering that the Chain Rule is necessary since
y is dependent on x.
Find
dy
dx
for the equation x2 + y + 3y 2 = 16
Find y 0 if (y 2 + 1)3 + xy = 3x2 − 2y. What is the slope of the tangent line at the point (2, 1)?
Find
dy
dx
if x2 − 5x4 y 2 = 4y 2
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Math 151, Benjamin
Aurispa
Find
dx
dy
if x2 − 5x4 y 2 = 4y 2
Find
dy
dx
where sin(x − y) + cos 2y = y cos x.
Example: Find an equation of the tangent line to the hyperbola
4
√
x2
y2
−
= 1 at the point (1, −3 5).
36
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Math 151, Benjamin
Aurispa
The curves x2 − y 2 = 5 and 4x2 + 9y 2 = 72 intersect at the point (3, −2). Show that the tangent lines to
the two curves at this point are orthogonal.
Definition: Two curves are called orthogonal if at every point of intersection, the tangent lines at those
points are orthogonal.
The above curves also intersect at (3, 2), (−3, 2), and (−3, −2). At each of these points, the tangent lines
are orthogonal, so these curves are orthogonal.
Two families of curves are orthogonal trajectories of each other if every curve of one family is orthogonal
to every curve in the other family.
Show that the families of curves y = ax3 and x2 + 3y 2 = b are orthogonal trajectories of each other.
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Math 151, Benjamin
Aurispa
3.7 Derivatives of Vector Functions
If r(t) =< x(t), y(t) > is a vector function, then r0 (t) =< x0 (t), y 0 (t) > if both of these derivatives exist.
Recall that the derivative at a, r0 (a), is a tangent vector to the curve when t = a. However, it also
represents the instantaneous velocity of an object with position function r(t). So, the tangent vector, (or
velocity vector), points in the direction of motion as t increases.
Example: Find the domain of the vector function r(t) =
√
t
, 3t − 5 .
t−6
Find a tangent vector to the curve at the point where t = 3.
Find parametric equations for the tangent line at this point.
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Math 151, Benjamin
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Example: Find a unit tangent vector to the curve r(t) =< t sin t, 4 − 2 cos 3t > at the point where t = π2 .
Find a unit tangent vector to the curve r(t) =< 5t2 + 1, 8t2 − t > at the point (6, 9).
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Math 151, Benjamin
Aurispa
Again, if r(t) =< x(t), y(t) > is vector function representing the position of a particle at time t, then the
tangent vector r0 (t) is the instantaneous velocity at time t and the instantaneous speed at time t is |r0 (t)|.
(Velocity = Vector; Speed = Scalar)
√
Example: A projectile is fired so that its position is given by the function r(t) =< (48 3)t, 48t − 16t2 >.
Find the velocity and speed at time t = 2.
What is the velocity when the projectile hits the ground?
To find the angle of intersection between two curves, find the angle between the tangent vectors at the point
of intersection.
√
Example: Find the angle of intersection of the curves r1 (t) =< 4 − t, t2 − 5 > and r2 (s) =< s − 1, s + 2 >
if it is known the curves intersect at the point (1, 4).
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