1 2 3 4 5 6 7 8 9 10

0:0010 lessons across the whole video52:50

You will be able to

Differentiate constants, monomials and polynomials using the power rule and constant multiple rule
Rewrite reciprocal and radical functions as rational exponents and differentiate them
Derive the derivative of x^2 from the limit definition and explain why it gives the slope of a tangent line
Estimate a tangent slope with secant lines and evaluate a derivative at a given x-value
Apply the product rule and quotient rule to differentiate products and quotients of functions
Simplify expressions such as distributed or divided polynomials before differentiating them

Good to know first

Exponent rules for multiplying, dividing and raising powers
Expanding binomials with FOIL and combining like terms
Slope of a line from two points
Basic understanding of limits and rational expressions

Syllabus

10 lessons and a final project

1
Derivatives of constants and the idea of slope
0:01–1:151 min of video
A derivative tells you the slope of a function at any x value. The derivative of any constant is zero, because a constant function is a flat horizontal line with slope zero.
3 build steps3 exercises3 questions
2
The Power Rule and the Constant Multiple Rule
1:15–8:007 min of video
You will learn the power rule, which finds the derivative of a power of x in one step, and the constant multiple rule, which lets you pull a constant out in front before differentiating. Together they let you differentiate any monomial such as 4x^7.
4 build steps3 exercises3 questions
3
The limit definition of the derivative: finding the derivative of x²
8:00–12:535 min of video
You will learn the limit definition of the derivative and use it to show, step by step, that the derivative of x² is 2x. The method works by comparing the function at x + h with the function at x, dividing by h, and then letting h approach zero.
2 formulas3 exercises3 questions
4
Tangent lines and secant line approximation
12:53–20:127 min of video
The derivative gives the slope of the tangent line at any point on a curve. A secant line through two nearby points gives an approximation of that slope, and the approximation gets better as the two points get closer together.
2 exercises3 questions
5
Differentiating Polynomials and Evaluating Slopes at a Point
20:12–26:306 min of video
You will learn to differentiate polynomial functions term by term using the power rule, the constant multiple rule, and the rule that constants vanish. You will then use the derivative to find the slope of the tangent line at a specific x-value.
3 build steps3 exercises3 questions
6
Negative exponents and reciprocal functions
26:30–30:444 min of video
You learn to turn a fraction like 1/x or 8/x^4 into a power with a negative exponent, so the power rule can differentiate it. Then you rewrite the answer back into fraction form.
2 exercises3 questions
7
Radical functions and rational exponents
30:44–36:005 min of video
You learn to rewrite a radical as a rational exponent so the power rule can be used to differentiate it. You then rewrite the answer back as a radical.
3 exercises3 questions
8
Simplifying before differentiating
36:00–41:065 min of video
Before taking a derivative, rewrite the function in a simpler polynomial form: distribute a monomial over a sum, expand a squared binomial, or divide each term of a fraction by the denominator. Then apply the power rule term by term.
3 exercises4 questions
9
Trigonometric derivatives and the product rule
41:06–49:178 min of video
You learn the derivatives of the six trig functions and how to differentiate a product of two functions, and then of three functions, using the product rule. The key idea is to take the derivative of one part at a time while the others stay the same.
4 build steps3 exercises4 questions
10
The Quotient Rule
49:17–52:504 min of video
The quotient rule gives the derivative of a fraction f/g as (g f' − f g') divided by g squared. You learn to name f and g, find their derivatives, plug them in, and simplify carefully.
1 formula3 exercises3 questions

Final project

Build a Mini Derivative Calculator and Verify It

Write a small Python program that differentiates polynomial-style terms with rational exponents and checks its answers against numerical secant slopes. It brings together the power rule, rewriting reciprocals and radicals, the limit definition, and tangent slopes.

Free preview · Lesson 1 in full

Derivatives of constants and the idea of slope

A derivative tells you the slope of a function at any x value. The derivative of any constant is zero, because a constant function is a flat horizontal line with slope zero.

0:01 to 1:15 in the video

Scratchpad

Draw and write next to the video.

The idea

The derivative of a function is another function that gives you the slope of the original function at any x value. Writing $f^{'} (x)$ (read "f prime of x") or $\frac{d}{d x} f (x)$ means you are differentiating the function with respect to x.

A constant is a number that never changes, such as 5, -7 or 8. The derivative of any constant is zero: $\frac{d}{d x} [c] = 0$ . For example, $\frac{d}{d x} (5) = 0$ and the derivative of -7 is also 0.

Why is it zero? Take $f (x) = 8$ . Its graph is a straight horizontal line at y = 8. The slope of a horizontal line is m = 0, no matter where you stand on it. Since the derivative gives the slope at every x, $f^{'} (x) = 0$ .

So the rule is simple: whenever you see a plain number with no x in it, its derivative is 0.

Key points

A derivative gives the slope of a function at any x value. 0:38
The derivative of any constant is zero. 0:12
The derivative of 5 is 0, and the derivative of -7 is also 0. 0:19
A horizontal line has slope zero, so the derivative of f(x) = 8 is f'(x) = 0. 1:06
D over DX means you are differentiating with respect to x. 0:58

Build along in the practice terminal

On paper, write d/dx [5] and find its derivative. Each constant gives 0.
Write f(x) = 8 and sketch a horizontal line at y = 8. Label its slope m = 0.
Conclude f'(x) = 0.

Practice

Calculate Solution checked

Find f'(x) if f(x) = -7.

Hint

Any plain number, positive or negative, has derivative zero.

Solution

f(x) = -7 is a constant function.
The derivative of any constant is 0.
So f'(x) = 0.

Calculate Solution checked

Find $\frac{d}{d x} (100)$ and $\frac{d}{d x} (- 3.5)$ .

Hint

Both are plain numbers with no x in them.

Solution

100 is a constant, so its derivative is 0.
-3.5 is also a constant, so its derivative is 0.

0 and 0

Explain Solution checked

In your own words, explain why the derivative of a constant is zero.

Hint

Think about what the graph of f(x) = 8 looks like and what its slope is.

Solution

A constant function's graph is a horizontal line.
A horizontal line has slope zero everywhere.
The derivative gives the slope at every x, so it is zero.

A constant's graph is a flat horizontal line with slope 0 at every point, so its derivative is 0.

On your scratchpad

Draw axes, then a horizontal line at y = 8. Write m = 0 beside it and write f'(x) = 0 underneath.

Quiz

What is the derivative of the constant 5?
1. 5
2. 1
3. 0
4. x
Show the answer
C. 0 · The derivative of any constant is zero.
0:19 watch the moment
What does a derivative give you?
1. The area under the graph
2. The slope of the function at any x value
3. The y-intercept
4. The x-intercept
Show the answer
B. The slope of the function at any x value · The derivative is a function that gives the slope at some x value.
0:38 watch the moment
Why is the derivative of f(x) = 8 equal to zero?
1. Because 8 is an even number
2. Because the graph is a horizontal line with slope 0
3. Because the graph is a vertical line
4. Because the graph rises 8 units for every 1 unit of x
Show the answer
B. Because the graph is a horizontal line with slope 0 · The graph is the horizontal line y = 8, whose slope is 0 everywhere, so f'(x) = 0.
1:06 watch the moment

Open lesson 1, with the video →

Made by Beyond from the video “Calculus 1 - Derivatives” by The Organic Chemistry Tutor. The video plays in YouTube's own player. Watch it on YouTube · All lessons

10 lessons and a final project

Derivatives of constants and the idea of slope

The Power Rule and the Constant Multiple Rule

The limit definition of the derivative: finding the derivative of x²

Tangent lines and secant line approximation

Differentiating Polynomials and Evaluating Slopes at a Point

Negative exponents and reciprocal functions

Radical functions and rational exponents

Simplifying before differentiating

Trigonometric derivatives and the product rule

The Quotient Rule

Build a Mini Derivative Calculator and Verify It

Derivatives of constants and the idea of slope

The idea

Key points

Build along in the practice terminal

Practice

On your scratchpad

Quiz