Worksheet: Implicit Differentiation

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

Q1:

Find the equation of the tangent to 9𝑦=7𝑥+9 that has slope 718.

  • A7𝑦+18𝑥+18=0
  • B9𝑦𝑥+18=0
  • C18𝑦7𝑥+1=0
  • D18𝑦7𝑥+18=0

Q2:

Find the equation of the tangent to the curve 𝑦=𝑥 at the point (1,1).

  • A𝑦=2𝑥3
  • B𝑦=2𝑥1
  • C𝑦=2𝑥+3
  • D𝑦=2𝑥+1
  • E𝑦=2𝑥1

Q3:

Given that 𝑥+9=2𝑥𝑦, find 𝑥𝑦𝑥+2𝑦𝑥dddd.

  • A4
  • B12
  • C1
  • D2

Q4:

Given that 𝑥+3𝑦=3, determine 𝑦 by implicit differentiation.

  • A𝑦=13𝑦
  • B𝑦=13𝑦
  • C𝑦=2𝑦13𝑦
  • D𝑦=2𝑥+39𝑦
  • E𝑦=𝑦+112𝑦

Q5:

Find the slope of the tangent to the curve 5𝑥2𝑦2𝑦𝑥=4 at the point (2,5).

  • A52
  • B56
  • C252
  • D53

Q6:

At a point on the curve 𝑥+3𝑥+𝑦+5𝑦+4=0 with 𝑥<0, 𝑦<0, the tangent makes an angle of 9𝜋4 with the positive 𝑥-axis. Find the equation of the tangent at that point.

  • A𝑥+𝑦+2=0
  • B𝑥+𝑦2=0
  • C𝑥+𝑦+4=0
  • D𝑥+𝑦4=0

Q7:

Find the equation of the tangent to the curve 2𝑦=87𝑥1 at the point (0,2).

  • A𝑦+𝑥72=0
  • B𝑦7𝑥2=0
  • C𝑦𝑥2=0
  • D𝑦+𝑥2=0

Q8:

Find the points on the curve 5𝑥8𝑥𝑦+4𝑦=4 at which the tangent is parallel to the 𝑦 axis.

  • A(1,1)
  • B(2,2)
  • C(2,2), (2,2)
  • D(1,1), (1,1)

Q9:

Find the points that lie on the curve 2𝑥𝑥𝑦+2𝑦48=0 at which the tangent is parallel to line 𝑦=𝑥.

  • A(4.76,2.85), (4.76,4.76)
  • B(4,4), (4,4)
  • C(4.76,2.85), (4.76,2.86)
  • D(4,4), (4,4)

Q10:

Find the equation of the tangent to the curve 5𝑥+4𝑦=19 at the point (3,4).

  • A16𝑦15𝑥109=0
  • B15𝑦16𝑥108=0
  • C16𝑦+15𝑥19=0
  • D16𝑦+15𝑥+19=0

Q11:

The tangent at (1,1) to the curve 𝑥9𝑥𝑦+8𝑦=0 makes a positive angle with the positive 𝑥-axis. Find this angle.

Q12:

Given that 2𝑥𝑥𝑦𝑦=1, determine 𝑦 by implicit differentiation.

  • A𝑦=9𝑥𝑦9𝑦(2𝑦+𝑥)
  • B𝑦=9𝑥9𝑦(2𝑦+𝑥)
  • C𝑦=9𝑥𝑦9𝑦(2𝑦+𝑥)
  • D𝑦=9𝑥9𝑦(2𝑦+𝑥)
  • E𝑦=7𝑥𝑦7𝑦(2𝑦+𝑥)

Q13:

Find the equations of the two tangents to the circle 𝑥+𝑦=125 that are inclined to the positive 𝑥-axis by an angle whose tangent is 2.

  • A𝑦+2𝑥+15=0, 𝑦+2𝑥15=0
  • B𝑦2𝑥25=0, 𝑦2𝑥+25=0
  • C2𝑦𝑥=0, 2𝑦𝑥=0
  • D2𝑦𝑥20=0, 2𝑦𝑥+20=0

Q14:

Find the equation of the tangent to the curve 4𝑥𝑦+3𝑥𝑦=1 at the point (1,1).

  • A5𝑥2+𝑦+32=0
  • B5𝑥2+𝑦72=0
  • C2𝑥5+𝑦35=0
  • D2𝑥5+𝑦75=0

Q15:

The tangent at (2,2) to the curve 𝑥+𝑥𝑦+5𝑥+5𝑦=0 makes a positive angle with the positive 𝑥-axis. Find this angle.

Q16:

Find the equation of the tangent to the curve sincos7𝑥=6𝑦 at the point 0,3𝜋4.

  • A7𝑦+6𝑥+21𝜋4=0
  • B6𝑦+7𝑥9𝜋2=0
  • C7𝑦6𝑥+21𝜋4=0
  • D6𝑦7𝑥9𝜋2=0

Q17:

Determine the points on a curve 𝑥+𝑦=45 at which the tangent to the curve is perpendicular to the straight line 𝑦=2𝑥+12.

  • A(3,6),(3,6)
  • B(3,6),(3,6)
  • C(6,3),(6,3)
  • D(6,3),(6,3)

Q18:

Find the equations of the normals to the curve 𝑥+3𝑥+𝑦2𝑦4=0 at the points on the 𝑥-axis.

  • A2𝑥+5𝑦2=0, 2𝑥+5𝑦8=0
  • B2𝑥+5𝑦+2=0, 2𝑥+5𝑦+8=0
  • C2𝑥+5𝑦+2=0, 2𝑥+5𝑦2=0
  • D5𝑥+2𝑦+5=0, 5𝑥+2𝑦+20=0

Q19:

Find the equation of the tangent to the curve 7𝑦+𝑦𝑥3𝑥=20 at the point (2,2).

  • A3𝑦+𝑥+8=0
  • B3𝑦𝑥+4=0
  • C𝑦+3𝑥+4=0
  • D𝑦+3𝑥+8=0

Q20:

Given that 8𝑥3𝑥5𝑦=0, find 𝑦𝑦𝑥+𝑦𝑥dddd.

  • A85
  • B8
  • C16
  • D85

Q21:

If 𝑥+𝑥𝑦+𝑦=1, find the value of 𝑦 at 𝑥=1.

Q22:

Find, for 0𝑥𝜋, the tangent to 9𝑦=(5𝑥+3𝑦)cos that has slope 512, giving your equation in terms of 𝜋.

  • A5𝑦12𝑥+6𝜋5=0
  • B12𝑦5𝑥+𝜋2=0
  • C5𝑦12𝑥+6𝜋5=0
  • D12𝑦+5𝑥𝜋2=0

Q23:

Find the equation of the tangent to 𝑥𝑦=15 that passes through (5,5).

  • A4𝑦𝑥15=0
  • B4𝑦+𝑥+25=0
  • C𝑦4𝑥+25=0
  • D𝑦+4𝑥15=0

Q24:

A tangent to 8𝑦=3𝑥+25 has slope 316. What is the equation of this line?

  • A3𝑥+16𝑦17=0
  • B𝑥8𝑦17=0
  • C3𝑥+16𝑦+34=0
  • D16𝑥3𝑦16=0

Q25:

Find dd𝑦𝑥, given that 6𝑥+6𝑦=25.

  • A𝑥2𝑦
  • B25𝑥2𝑦
  • C25𝑥2𝑦
  • D75𝑥𝑦

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