This is for Basic Calculus

1. THE DERIVATIVE AS
THE SLOPE OF THE
TANGENT LINE

2. WHAT IS A DERIVATIVE?
•The rate of change of a
function
•the slope of the line
tangent to the curve

3. THE TANGENT LINE
single point
of intersection

4. SLOPE OF A SECANT LINE
x
x+ Δx
f(x+ Δx)
f(x)
f(x+ Δx) - f(x)
x+ Δx - x
Difference
Qoutient

5. SLOPE OF A (CLOSER)
SECANT LINE
x
f(x)
x + Δx x + Δx

6. CLOSER AND CLOSER…
x

7. WATCH THE SLOPE...

8. WATCH WHAT X DOES...
a
x

9. THE SLOPE OF THE SECANT LINE GETS
CLOSER AND CLOSER TO THE SLOPE
OF THE TANGENT LINE...

10. AS THE VALUES OF X GET CLOSER
AND CLOSER TO A!
x
x+ Δx

11. The slope of the secant lines
gets closer
to the slope of the tangent line...
...as the values of Δx
approches to 0
Translates to….

12. lim
0
Δx
f(x + Δx) - f(x)
Δx
Equation for the slope
Which gives us the the exact slope
of the line tangent to the curve at a!
as x goes to a

13. COMPARISON
Difference Quotient
•
• Slope of secant
• Average rate of change
• Average velocity
13
Derivative
• Slope of tangent
• Instantaneous rate of
change
• Instantaneous velocity

14. DERIVATIVES
• The derivative of the function y = f (x) may
be expressed as …
'( )
f x
'
y
dy
dx
Prime notation
Leibniz notation
“f prime of x”
“y prime”
“the derivative of y with respect to x”

15. The limit definition to find the
derivative of:

16. Use the limit definition to find the
derivative of:

18. Find the slope of the tangent line to each
curve when x has the indicated value.

19. Find the equation of the tangent line to the
functions at a given point.