1. Techniques of Differentiation
The Product and Quotient Rules
The Chain Rule
Derivatives of Logarithmic and Exponential Functions
Implicit Differentiation

2. The Product Rule
The Quotient Rule

3. The Product Rule
Ex.
Derivative of Second
Derivative of first

4. The Quotient Rule Ex. Derivative of denominator
Derivative of numerator

5. Compute the Derivative
Ex.
= –10

6. The Chain Rule
If f is a differentiable function of u and u is a differentiable function of x, then the composite f (u) is a differentiable function of x, and
The derivative of a f (quantity) is the derivative of f evaluated at the quantity, times the derivative of the quantity.

7. Generalized Power Rule
Ex.

8. The Chain Rule
Ex.

9. Chain Rule in Differential Notation
If y is a differentiable function of u and u is a differentiable function of x, then

10. Chain Rule Example
Ex.
Sub in for u

11. Differentiation of Logarithmic Functions
Derivative of the Natural Logarithm
Generalized Rule for Natural Logarithm Functions
If u is a differentiable function, then

12. Examples Ex. Find the derivative of
Ex. Find an equation of the tangent line to the graph of
Slope:
Equation:

13. Differentiation of Logarithmic Functions
Derivative of a Logarithmic Function:
Generalized Rule for Logarithm Functions
If u is a differentiable function, then

14. Differentiation of Logarithmic Functions
Ex.

15. Derivative of Logarithms of Absolute Values

16. Derivative of Logarithms of Absolute Values
Ex.
Ex.

17. Differentiation of Exponential Functions
Derivative of ex:
Generalized Rule for eu:
If u is a differentiable function, then

18. Derivatives of Exponential Functions
Ex. Find the derivative of
Ex. Find the derivative of

19. Differentiation of Exponential Functions
Derivative of bx:
Generalized Rule for bu:
If u is a differentiable function, then

20. Derivatives of Exponential Functions
Ex. Find the derivative of

21. Implicit Differentiation
y is explicitly a function of x.
y is implicitly a function of x.

22. Implicit Differentiation (cont.)
To differentiate the implicit case we use the chain rule where y is a function of x:
Solve for

23. Tangent Line to Implicit Curve
Ex. Find the equation of the tangent line to the curve
at the point (2, 1).

24. Logarithmic Differentiation
Ex. Use logarithmic differentiation to find the derivative of
Apply ln
Properties of ln
Differentiate
Solve