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Chapter 3 | Derivatives
Example 3.69 Using Implicit Differentiation and the Product Rule
Assuming that y is defined implicitly by the equation x 3 sin y + y =4 x +3, find dy dx .
Solution
⎛ ⎝ x 3 sin y + y ⎞
d dx
⎠ = d
dx (4
x +3)
Step 1: Differentiate both sides of the equation. Step 1.1: Apply the sum rule on the left. On the right, d dx (4 x +3) =4.
⎛ ⎝ x 3 sin y
⎞ ⎠ + d
d dx
dx (
y ) = 4
Step 1.2: Use the product rule to find d dx ⎛ ⎝ x 3 sin y ⎞ ⎠ .Observe that d dx ( y ) = dy dx .
⎛ ⎝ d
⎞ ⎠ +
⎛ ⎝ x 3
⎞ ⎠ · sin y + d dx ⎛
dy dx = 4
⎞ ⎠ · x 3
⎝ sin y
dx
Step 1.3: We know d dx ⎛ ⎝ x 3 ⎞ chain rule to obtain d dx ⎛ ⎝ sin y
⎠ =3 x 2 .Use the
⎛ ⎝ cos y
⎞ ⎠ · x 3 +
dy dx
dy dx = 4
3 x 2 sin y +
dy dx .
⎞ ⎠ =cos y
Step 2: Keep all terms containing dy left. Move all other terms to the right.
dx on the
dy dx +
dy dx = 4−3
x 3 cos y
x 2 sin y
⎛ ⎝ x 3 cos y +1 ⎞
dy dx
dy dx on the left.
⎠ = 4−3 x 2 sin y
Step 3: Factor out
dy dx by dividing both sides of
Step 4: Solve for
x 2 sin y x 3 cos y +1
dy dx = 4−3
the equation byx 3 cos y +1.
Example 3.70 Using Implicit Differentiation to Find a Second Derivative
d 2 y dx 2
if x 2 + y 2 =25.
Find
Solution In Example 3.68 , we showed that
dy dx = −
x y . We can take the derivative of both sides of this equation to find
d 2 y dx 2
.
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