MATH&151 Final Project Fundamentals of Derivatives.pdf

Discusses the basis of modern day derivatives in calculus.

1. THE FUNDAMENTALS
OF DERIVATIVES
MATH&151 Final Project by
Christina Nguyen

2. THE HISTORY OF DERIVATIVES:THE INVENTORS OF DERIVATIVES
AND MODERN DIFFERENTIATION
Sir Issac Newton Gottfried Wilhelm Leibniz

3. THE HISTORY OF DERIVATIVES:A TIME LINE
Ancient and Medieval
Mathematicians
Archimedes and other mathematicians during this period
used methods that foreshadowed modern-day calculus, such
as approximating areas with infinitesimally sub divisions.
Pierre de Fermat (1607-1665)
Developed a method to find tangents to curves that was very
similar to taking the limit of a small change.
Isaac Barrow (1630-1677)
A teacher of Newton,Barrow further developed ideas
related to derivatives, which later served as the foundation
for Newton's later work.
Newton and Leibniz (17th
Century)
Both Newton and Leibniz independently developed the
concept of derivatives that would now serve as the basis of
modern-day derivatives in calculus.
Modern Era of Derivatives
Derivatives in calculus ultimately allow mathematicians to
solve problems related to the rate of change, curvature, and
optimization with greater accuracy and integration.

4. THE FUNDAMENTALS OF DERIVATIVES:AN OVERVIEW
The derivative is a function that measures the
instantaneous rate of changeof another function
at a given point. It explains how fast a function
is changing at a specific moment in time.
Ultimately, it represents the slope of a tangent
line to the function's graph at the particular
point.
Notation: It can be written as f '(x),d/dx,
ΔyΔx, etc.
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Using the equation below, we can
calculate the derivative of a function
where f is a function and x is a value in
the function's domain, provided the limit
exists.

5. THE FUNDAMENTALS OF DERIVATIVES: AN OVERVIEW
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Definition of a Derivative. CUEMATH, n.d.

6. HOW TO SOLVE FOR THE DERIVATIVE AT A POINT USING THE
FORMULA
Find the derivative of y = x³ - 5, at x = 2.
First, find the equation of the tangent slope line of the function at x = 2.
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f (x + h) - f (x)
f ' (x) = lim
h→0
_____________
h
Plug (x + h) and (x) into
the function y = x³ - 5.
__________________
f ' (x) = lim
h→0 h
[(x + h)³- 5] - [x³ - 5]

7. HOW TO SOLVE FOR THE DERIVATIVE AT A POINT USING THE
FORMULA
Find the derivative of y = x³ - 5, at x = 2.
Using arithmetic, expand the equation.
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Multiple (x + h) three times by itself and
multiply the negative constant by (x³ - 5).
__________________
f ' (x) = lim
h→0 h
[(x + h)³- 5] - [x³ - 5]
____________________________
f ' (x) = lim
h→0 h
x³ + 3x²h + 3xh² + h³ - 5 - x³ + 5

8. HOW TO SOLVE FOR THE DERIVATIVE AT A POINT USING THE
FORMULA
Find the derivative of y = x³ - 5, at x = 2.
Cancel terms as necessary.
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Cancel out h terms.
____________________________
f ' (x) = lim
h→0 h
x³ + 3x²h + 3xh² + h³ - 5- x³ + 5
_______________
f ' (x) = lim
h→0 h
3x²h + 3xh² + h³
h h²

9. HOW TO SOLVE FOR THE DERIVATIVE AT A POINT USING THE
FORMULA
Find the derivative of y = x³ - 5, at x = 2.
Now take the limit as h approaches 0
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f ' (x) = lim
h→0
3x² + 3xh + h²
→0→0
f ' (x) = 3x²Equation for the tangent line slope

10. HOW TO SOLVE FOR THE DERIVATIVE AT A POINT USING THE
FORMULA
Find the derivative of y = x³ - 5, at x = 2.
Now we can plug in x = 2 into the tangent line slope equation
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f ' (2) = 3(2)²
= 12Tangent line slope, also known as the derivative

11. HOW TO SOLVE FOR THE DERIVATIVE AT A POINT USING THE
FORMULA
Find the derivative of y = x³ - 5, at x = 2.
Now that we have found the tangent line, we can use the point-slope intercept formula to find
where the line touches the curve at x = 2.
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f(2) = 2³ - 5
f(2) = 3
To find the point, plug in x = 2 to solve for y.
The point is at (2,3).
y - y₁ = m(x - x₁)Point-slope intercept form.
y - 3= 12(x - 2)Plug in the x and y values and slope.

12. HOW TO SOLVE FOR THE DERIVATIVE AT A POINT USING THE
FORMULA
Find the derivative of y = x³ - 5, at x = 2.
Solve using arithmetic.
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y - 3= 12(x - 2)
y - 3= 12x - 24
y= 12x - 21

13. BASIC DERIVATIVE RULES
•Constant Rule
•Power Rule
•Product Rule
•Quotient Rule
•Chain Rule
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14. FUNDAMENTALS OF DERIVATIVES: THE CONSTANT RULE
What is the Constant Rule?
•The derivative of a constant is 0
Example of the Constant Rule
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y = 3
f ' (x) = 0

15. FUNDAMENTALS OF DERIVATIVES: THE POWER RULE
What is the Power Rule?
•A quick way to find the derivative of a
function
•Works when each term in the expression is a
variable
How to Solve Derivatives Using the Power Rule
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16. FUNDAMENTALS OF DERIVATIVES: THE PRODUCT RULE
What is the Product Rule?
•Used to find the derivative of products of
two or more differentiable functions
How to Solve Derivatives Using the Product Rule
y = 2x²y
f'(x)= 4x(y) + 2x²(1)
f'(x)= 4xy + 2x²
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+

17. _______f'(x) = -x⁴ -3x²
______
FUNDAMENTALS OF DERIVATIVES: THE QUOTIENT RULE
What is the Quotient Rule?
•Used to find the derivative of a function that
is a ratio of two differentiable functions
How to Solve Derivatives Using the Quotient Rule
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f'(x) g(x) - g '(x) f(x)
_________________
[g(x)]²
y = x² + 1
x³
f'(x)= (2x)(x³) - 3x²(x² + 1)
________________
[x³]²
x⁴
______= -x²-3
x⁶

18. FUNDAMENTALS OF DERIVATIVES: THE CHAIN RULE
What is the Chain Rule?
•Used to find the derivative of a composite
function
•For example, if y = f(g(x))
How to Solve Derivatives Using the Chain Rule
y = (3x² + 4)³
f'(x) = 3(3x² + 4)² (6x)
f'(x) = 18x(3x² + 4)²
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19. REAL-LIFE APPLICATIONS OF DERIVATIVES: RELATED RATES
•Finding the rate at which a quantity
changes by relating it to other changing
quantities
•Implicit differentiation
•Taking the derivative with respect to
time
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20. REAL-LIFE APPLICATIONS OF DERIVATIVES: OPTIMIZATION
•Finding the largest possible value or
the smallest possible value that a
function can take
•Maximums and minimums
•First derivative and second derivative
test
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21. THE END
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22. BIBLIOGRAPHY
Calcworkshop. (n.d.). Save Your Grade with the Step-By-Step Online Courses That Take the Mystery Out of
Learning Math. Calcworkshop - Giving You Confidence In Math. https://calcworkshop.com/
CUEMATH. (n.d.). Derivatives. Derivatives - Calculus, Meaning, Interpretation.
https://www.cuemath.com/calculus/derivatives/
Oxford College. (n.d.). The Department of Mathematics and Computer Science. The Definition of the Derivative.
https://mathcenter.oxford.emory.edu/site/math111/defOfDerivative/
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23. QUESTIONS?
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