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.

  • A 7 𝑦 + 1 8 𝑥 + 1 8 = 0
  • B 9 𝑦 𝑥 + 1 8 = 0
  • C 1 8 𝑦 7 𝑥 + 1 = 0
  • D 1 8 𝑦 7 𝑥 + 1 8 = 0

Q2:

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

  • A 𝑦 = 2 𝑥 3
  • B 𝑦 = 2 𝑥 1
  • C 𝑦 = 2 𝑥 + 3
  • D 𝑦 = 2 𝑥 + 1
  • E 𝑦 = 2 𝑥 1

Q3:

Given that 𝑥+9=2𝑥𝑦, find 𝑥𝑦𝑥+2𝑦𝑥dddd.

  • A 4
  • B 1 2
  • C 1
  • D2

Q4:

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

  • A 𝑦 = 1 3 𝑦
  • B 𝑦 = 1 3 𝑦
  • C 𝑦 = 2 𝑦 1 3 𝑦
  • D 𝑦 = 2 𝑥 + 3 9 𝑦
  • E 𝑦 = 𝑦 + 1 1 2 𝑦

Q5:

Do the curves 9𝑦8𝑦=6𝑥 and 5𝑥3𝑦=4𝑥 intersect orthogonally at the origin?

  • Ayes
  • Bno

Q6:

Determine the equation of the tangent to the curve 6𝑥+𝑥𝑦5𝑦=0 at the point (2,4).

  • A 1 1 𝑥 3 + 𝑦 + 1 0 3 = 0
  • B 1 1 𝑥 3 + 𝑦 + 1 0 3 = 0
  • C 1 1 𝑥 3 + 𝑦 3 4 3 = 0
  • D 3 𝑥 1 1 + 𝑦 5 0 1 1 = 0

Q7:

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

  • A 5 2
  • B 5 6
  • C 2 5 2
  • D 5 3

Q8:

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.

  • A 𝑥 + 𝑦 + 2 = 0
  • B 𝑥 + 𝑦 2 = 0
  • C 𝑥 + 𝑦 + 4 = 0
  • D 𝑥 + 𝑦 4 = 0

Q9:

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

  • A 𝑦 + 𝑥 7 2 = 0
  • B 𝑦 7 𝑥 2 = 0
  • C 𝑦 𝑥 2 = 0
  • D 𝑦 + 𝑥 2 = 0

Q10:

Find the equation of the tangent to the curve 9𝑥6𝑥+6𝑥𝑦𝑦+2=0 at the point (0,1).

  • A 𝑦 + 𝑥 2 1 = 0
  • B 𝑦 3 𝑥 1 = 0
  • C 𝑦 2 𝑥 1 = 0
  • D 𝑦 + 𝑥 3 1 = 0

Q11:

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 )

Q12:

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

  • A ( 4 . 7 6 , 2 . 8 5 ) , ( 4 . 7 6 , 4 . 7 6 )
  • B ( 4 , 4 ) , ( 4 , 4 )
  • C ( 4 . 7 6 , 2 . 8 5 ) , ( 4 . 7 6 , 2 . 8 6 )
  • D ( 4 , 4 ) , ( 4 , 4 )

Q13:

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

  • A 1 6 𝑦 1 5 𝑥 1 0 9 = 0
  • B 1 5 𝑦 1 6 𝑥 1 0 8 = 0
  • C 1 6 𝑦 + 1 5 𝑥 1 9 = 0
  • D 1 6 𝑦 + 1 5 𝑥 + 1 9 = 0

Q14:

The tangent at (2,1) to the curve 6𝑥+4𝑥𝑦𝑦=17 makes a positive angle with the positive 𝑥-axis. Find this angle to the nearest minute.

  • A 5 3 8
  • B 7 3 1 8
  • C 4 8 4 9
  • D 7 0 4 3

Q15:

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

Q16:

A tangent to 𝑥+𝑦=72 forms an isosceles triangle when taken with the positive 𝑥- and 𝑦-axes. What is the equation of this tangent?

  • A 𝑥 + 𝑦 1 2 = 0
  • B 𝑥 + 𝑦 1 2 = 0
  • C 𝑥 + 𝑦 = 0
  • D 𝑥 + 𝑦 = 0

Q17:

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

  • A 𝑦 = 9 𝑥 𝑦 9 𝑦 ( 2 𝑦 + 𝑥 )
  • B 𝑦 = 9 𝑥 9 𝑦 ( 2 𝑦 + 𝑥 )
  • C 𝑦 = 9 𝑥 𝑦 9 𝑦 ( 2 𝑦 + 𝑥 )
  • D 𝑦 = 9 𝑥 9 𝑦 ( 2 𝑦 + 𝑥 )
  • E 𝑦 = 7 𝑥 𝑦 7 𝑦 ( 2 𝑦 + 𝑥 )

Q18:

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.

  • A 𝑦 + 2 𝑥 + 1 5 = 0 , 𝑦 + 2 𝑥 1 5 = 0
  • B 𝑦 2 𝑥 2 5 = 0 , 𝑦 2 𝑥 + 2 5 = 0
  • C 2 𝑦 𝑥 = 0 , 2 𝑦 𝑥 = 0
  • D 2 𝑦 𝑥 2 0 = 0 , 2 𝑦 𝑥 + 2 0 = 0

Q19:

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

  • A 5 𝑥 2 + 𝑦 + 3 2 = 0
  • B 5 𝑥 2 + 𝑦 7 2 = 0
  • C 2 𝑥 5 + 𝑦 3 5 = 0
  • D 2 𝑥 5 + 𝑦 7 5 = 0

Q20:

The point (5,2) lies on the curve 𝑥+𝑦3𝑘𝑥+7=0. Find 𝑘 and the equation of the tangent to the curve at this point.

  • A 𝑘 = 1 2 5 , equation of the tangent: 43𝑥10+𝑦+472=0
  • B 𝑘 = 1 2 5 , equation of the tangent: 7𝑥10+𝑦+112=0
  • C 𝑘 = 1 2 5 , equation of the tangent: 7𝑥10+𝑦+32=0
  • D 𝑘 = 1 2 5 , equation of the tangent: 7𝑥10+𝑦+112=0

Q21:

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.

  • A 3 0
  • B 4 5
  • C 9 0
  • D 6 0
  • E 1 3 5

Q22:

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

  • A 7 𝑦 + 6 𝑥 + 2 1 𝜋 4 = 0
  • B 6 𝑦 + 7 𝑥 9 𝜋 2 = 0
  • C 7 𝑦 6 𝑥 + 2 1 𝜋 4 = 0
  • D 6 𝑦 7 𝑥 9 𝜋 2 = 0

Q23:

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 )

Q24:

Find the equation of the normal to the curve 𝑦=52𝑥1 at the point (3,1).

  • A 𝑦 5 𝑥 + 1 4 = 0
  • B 𝑦 + 𝑥 1 0 1 3 1 0 = 0
  • C 𝑦 1 0 𝑥 + 2 9 = 0
  • D 𝑦 + 𝑥 5 8 5 = 0

Q25:

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

  • A 2 𝑥 + 5 𝑦 2 = 0 , 2 𝑥 + 5 𝑦 8 = 0
  • B 2 𝑥 + 5 𝑦 + 2 = 0 , 2 𝑥 + 5 𝑦 + 8 = 0
  • C 2 𝑥 + 5 𝑦 + 2 = 0 , 2 𝑥 + 5 𝑦 2 = 0
  • D 5 𝑥 + 2 𝑦 + 5 = 0 , 5 𝑥 + 2 𝑦 + 2 0 = 0

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