The portal has been deactivated. Please contact your portal admin.

Lesson Worksheet: Implicit Differentiation Mathematics • Higher Education

In this worksheet, we will practice using implicit differentiation to differentiate functions defined implicitly.

Q1:

Given that 4π‘₯βˆ’2𝑦+18=0, find dd𝑦π‘₯.

  • A2π‘₯π‘¦οŠ¨
  • B2π‘₯3π‘¦οŠ¨οŠ¨
  • C2π‘₯π‘¦οŠ¨οŠ¨
  • D0

Q2:

Find the slope of the tangent to the curve π‘₯𝑦=βˆ’16 at (4,βˆ’1).

  • A12
  • Bβˆ’16
  • C16
  • D112

Q3:

Find the equation of the tangent to 9𝑦=βˆ’7π‘₯+9 that has slope 718.

  • Aβˆ’7𝑦+18π‘₯+18=0
  • B9π‘¦βˆ’π‘₯+18=0
  • Cβˆ’18π‘¦βˆ’7π‘₯+1=0
  • D18π‘¦βˆ’7π‘₯+18=0

Q4:

Given that π‘₯+3𝑦=3, determine 𝑦′′ by implicit differentiation.

  • A𝑦′′=13π‘¦οŠ©
  • B𝑦′′=βˆ’13π‘¦οŠ©
  • C𝑦′′=2π‘¦βˆ’13π‘¦οŠ¨οŠ©
  • D𝑦′′=βˆ’2π‘₯+39π‘¦οŠ¨οŠ¨
  • E𝑦′′=βˆ’π‘¦+112π‘¦οŠ¨οŠ¨

Q5:

Find the slope of the tangent to the curve 5π‘₯2π‘¦βˆ’2𝑦π‘₯=βˆ’4 at the point (2,5).

  • A52
  • B56
  • C252
  • D53

Q6:

Find, for 0≀π‘₯β‰€πœ‹, the tangent to 9𝑦=(5π‘₯+3𝑦)cos that has slope βˆ’512, giving your equation in terms of πœ‹.

  • A5π‘¦βˆ’12π‘₯+6πœ‹5=0
  • B12π‘¦βˆ’5π‘₯+πœ‹2=0
  • Cβˆ’5π‘¦βˆ’12π‘₯+6πœ‹5=0
  • D12𝑦+5π‘₯βˆ’πœ‹2=0

Q7:

Given that sincos𝑦+2π‘₯=5, determine 𝑦′′ by implicit differentiation.

  • A𝑦′′=2π‘₯π‘¦βˆ’2π‘₯𝑦𝑦sinsincoscoscos
  • B𝑦′′=βˆ’4π‘₯𝑦+2π‘₯𝑦𝑦sinsincoscoscos
  • C𝑦′′=4π‘₯𝑦+2π‘₯𝑦𝑦sinsincoscoscos
  • D𝑦′′=2π‘₯𝑦+2π‘₯𝑦𝑦sinsincoscoscos
  • E𝑦′′=βˆ’4π‘₯𝑦+2π‘₯𝑦𝑦sinsincoscoscos

Q8:

Consider the equation π‘₯+𝑦=1.

Using implicit differentiation, find an expression for dd𝑦π‘₯ in terms of π‘₯ and 𝑦.

  • Add𝑦π‘₯=1βˆ’2π‘₯2𝑦
  • Bdd𝑦π‘₯=βˆ’π‘₯𝑦
  • Cdd𝑦π‘₯=π‘₯𝑦
  • Ddd𝑦π‘₯=βˆ’1βˆ’2π‘₯2𝑦

For the semicircle where 𝑦β‰₯0, express 𝑦 explicitly in terms of π‘₯; then, differentiate this expression to get an expression for dd𝑦π‘₯ in terms of π‘₯.

  • Add𝑦π‘₯=π‘₯√1βˆ’π‘₯
  • Bdd𝑦π‘₯=βˆ’12√1βˆ’π‘₯
  • Cdd𝑦π‘₯=βˆ’π‘₯√1βˆ’π‘₯
  • Ddd𝑦π‘₯=12√1βˆ’π‘₯

Q9:

The equation π‘¦βˆ’24π‘₯+24π‘₯=0 describes a curve in the plane.

Find the coordinates of two points on this curve, where π‘₯=βˆ’12.

  • Aο€Όβˆ’12,9 and ο€Όβˆ’12,βˆ’9
  • BThe curve does not pass through the point π‘₯=βˆ’12.
  • Cο€Όβˆ’12,√3 and ο€Όβˆ’12,βˆ’βˆš3
  • Dο€Όβˆ’12,√15 and ο€Όβˆ’12,βˆ’βˆš15
  • Eο€Όβˆ’12,3 and ο€Όβˆ’12,βˆ’3

Determine the equation of the tangent at the points where π‘₯=βˆ’12 and the 𝑦-coordinate is positive.

  • A𝑦=52βˆ’π‘₯
  • B𝑦=72+π‘₯
  • C𝑦=172βˆ’π‘₯
  • D𝑦=βˆ’1+2√32βˆ’π‘₯
  • E𝑦=βˆ’52+π‘₯

Find the coordinates of another point, if it exists, at which the tangent meets the curve.

  • AThey do not meet at any other point.
  • Bο€Ό2524,3524
  • Cο€Ό2524,βˆ’3524
  • Dο€Ό1,32
  • Eο€Ό3524,2524

Q10:

Find ddοŠ©οŠ©π‘¦π‘₯, given that 6π‘₯+6𝑦=25.

  • Aβˆ’π‘₯2π‘¦οŠ«
  • Bβˆ’25π‘₯2π‘¦οŠ¬
  • Cβˆ’25π‘₯2π‘¦οŠ«
  • Dβˆ’75π‘₯π‘¦οŠ«

This lesson includes 110 additional questions and 843 additional question variations for subscribers.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.