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Lesson Worksheet: Implicit Differentiation Mathematics • Higher Education

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


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

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


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

  • A12
  • Bβˆ’16
  • C16
  • D112


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


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¨


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

  • A52
  • B56
  • C252
  • D53


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


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


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‘₯


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


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.

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