implicit

1. Good
Afternoon! 

2. Dear Lord and Father of all,
Thank you for today.
Thank you for ways in which you provide for us all. For
Your protection and love, we thank you. Help us to focus
our hearts and minds now on what we are about to learn.
Inspire us by Your Holy Spirit as we listen and write.
Guide us by your eternal light as we discover the world
around us.
We ask this in the name of Jesus. Amen.
OPENING PRAYER

3. ATTENDANCE

4. THE CHAIN RULE

5. What are the different rules in
derivatives?

6. Example:
𝑓 𝑥 = (3𝑥 − 2)(5𝑥 + 4)4

7. ADVANCED
DIFFERENTIATIO
N RULES

8. IMPLICIT DIFFERENTIATION
A function is written in explicit form
when it is expressed as y = f(x).
However, there are tomes when
mathematical functions are written in a
more complicated form, wherein it is
difficult to express y explicitly in terms of
x. Such functions are expressed in

9. The implicit differentiation process
1. Differentiate both sides of the equation
with respect to x.
2. Combine all terms containing
𝑑𝑦
𝑑𝑥
on the
left side of the equation and all the other
terms on the right side.
3. On the left side of the equation , factor out
𝑑𝑦
𝑑𝑥
.

10. Find
𝑑𝑦
𝑑𝑥
given that 𝑦3
+ 𝑦 − 3𝑥2
= 2𝑥 + 5
1. Differentiate both sides of the equation
with respect to x.
𝑑
𝑑𝑥
𝑦3
+ 𝑦 − 3𝑥2
=
𝑑
𝑑𝑥
(2𝑥 + 5)
3𝑦2
+
𝑑𝑦
𝑑𝑥
− 6𝑥 = 2

11. Find
𝑑𝑦
𝑑𝑥
given that 𝑦3
+ 𝑦 − 3𝑥2
= 2𝑥 + 5
Combine all terms containing
𝑑𝑦
𝑑𝑥
on the left
side of the
3𝑦2
+
𝑑𝑦
𝑑𝑥
= 6𝑥 + 2

12. Find
𝑑𝑦
𝑑𝑥
given that 𝑦3
+ 𝑦 − 3𝑥2
= 2𝑥 + 5
factor out
𝑑𝑦
𝑑𝑥
.
𝑑𝑦
𝑑𝑥
(3𝑦2
+ 1) = 6𝑥 + 2

13. Find
𝑑𝑦
𝑑𝑥
given that 𝑦3
+ 𝑦 − 3𝑥2
= 2𝑥 + 5
Isolate
𝑑𝑦
𝑑𝑥
by dividing out the outer factor on
the left side.
𝑑𝑦
𝑑𝑥
=
6𝑥 + 2
3𝑦2 + 1

14. 𝑥2
+ 𝑦2
= 9
𝑑
𝑑𝑥
𝑥2
+
𝑑
𝑑𝑥
𝑦2
=
𝑑𝑦
𝑑𝑥
9
2x + 2𝑦
𝑑
𝑑𝑥
= 0

15. 3𝑥2
+ 𝑦2
= 9
𝑑
𝑑𝑥
3𝑥2
+
𝑑
𝑑𝑥
𝑦2
=
𝑑𝑦
𝑑𝑥
9
6x + 2𝑦
𝑑
𝑑𝑥
= 0

16. 𝑦 − 𝑥2
+ 2𝑥 = 0
𝑑
𝑑𝑥
𝑦 −
𝑑
𝑑𝑥
𝑥2
+
𝑑
𝑑𝑥
2𝑥 =
𝑑
𝑑𝑥
9
𝑑𝑦
𝑑𝑥
− 2𝑥 + 2 = 0

17. 𝑥2
+ 𝑦2
+ 2𝑦 = 4
𝑑
𝑑𝑥
𝑥2
+
𝑑
𝑑𝑥
𝑦2
+
𝑑
𝑑𝑥
2𝑦 =
𝑑
𝑑𝑥
4

18. Find the implicit differentiation
for each item.
1.𝑥3
+ 5𝑦3
= 2
2.−4𝑦3
+ 1 = 4𝑥
3.−5𝑦3
+ 3𝑦2
= 𝑥
4.𝑥2
+ 5𝑦2
= 3𝑦3
5.𝑥2
− 2𝑥𝑦 + 𝑦3
= 𝑐

19. Find the slope of the tangent
line to the curved described by
𝑥3
= 1 𝑎𝑡 (2,8)
1. Find the derivative of f(x)
2. Find the slope of the tangent
line to the curve at the point by
substituting to f(x)

20. Find the equation of the
tangent line to the curved f x =
𝑥3
+ 3𝑥2
− 4𝑥 − 5 𝑎𝑡 (1, −5)
1. Find the derivative of f(x)
2. Find the slope of the tangent line to the
curve at the point by substituting to f(x)
3. Use the point slope form of the line to find
the equation of the tangent line.

21. Find the equation of the
tangent line to the curved f x =
𝑥2
− 4𝑥 − 5 𝑎𝑡 (1, −5)
1. Find the derivative of f(x)
2. Find the slope of the tangent line to the
curve at the point by substituting to f(x)
3. Use the point slope form of the line to find
the equation of the tangent line.

22. 𝑥3
+ 𝑦3
= 9 𝑎𝑡 (1, 2)

23. 𝑥3
+ 𝑦3
= 9 𝑎𝑡 (1, 2)

24. 2𝑥3
= 2𝑦2
+ 5

25. Dear Lord
Thank you that you promise us that when two or
more come together in Your name
You are with us.
Thank you, Lord, that you have been with
us throughout this lesson.
And that you are with us right now.
Inspire us as we leave this place to love and
serve You always. Amen.
CLOSING PRAYER