MAT 271 Lab # 4
Name: __________________________________
Implicit and Logarithmic Differentiation
Show all work for full credit. Clearly mark answers. Include units where appropriate. Do not round any
numerical answers.
Concept Questions
How do you know when to use Implicit Differentiation versus Explicit Differentiation?
How do you know when you must use Logarithmic Differentiation?
Implicit Differentiation
Determine
dy
for each of the following.
dx
1. 3x2 − 6 y5 + 3 = ln( x)
2. 2 x2 y8 − 6 y5 + 3x2 = 5
3. sec( xy) + sin( y) = tan( x)
4. ln( y ) − 3cos( x + y ) +
x
=6
y
5. 12 x 2 − y 2 − 3e 4 y −5 x 7 + 14 x cos( y ) = e
Logarithmic Differentiation
Determine
dy
for each of the following on the intervals in which each is differentiable.
dx
6. y = x x
7. y = (3x)(ln( x ))
8. y = ( sin( x) )
( cos( x ) )
9. y = (ln(5x + 1))(3 x)
10. y = (tan( x))(ln( x ))
More Practice
d2y
11. Determine
for x2 + 9 y 2 = 1 in terms of x and y only.
2
dx
12. Determine the equation of the tangent line for ( x2 + y 2 )5 = 32 x2 y 2 at the point (1, −1) .
13. Determine the equation of the tangent line for tan( xy) = sin( y) at the point (0,  ) .