You will be able to
- Differentiate constants, monomials, polynomials and negative-exponent terms using the power and constant multiple rules
- Rewrite radical and rational expressions with exponents and differentiate them
- Derive the derivative of x^2 from the limit definition and explain it as a tangent-line slope
- Use the product and quotient rules to differentiate products and quotients of functions
- Apply the derivatives of sin, cos, tan, sec and cot and evaluate a derivative to find a tangent slope
Good to know first
- Exponent rules for multiplying, dividing and negative exponents
- Basic function notation and evaluating functions at a point
- Slope of a line and the slope formula between two points
- Expanding binomials with FOIL and combining like terms
Syllabus
10 lessons and a final project
- 1
Derivatives of constants and the power rule Free
You will learn that the derivative of a constant is zero, and how the power rule and the constant multiple rule let you differentiate monomials like x^n and c·x^n. You will practise on several examples and check each answer step by step.
4 build steps3 exercises4 questions - 2
The limit definition of the derivative: proving the derivative of x² is 2x Beyond Pass
You will learn the limit definition of the derivative, f'(x) = lim h→0 [f(x+h) − f(x)]/h, and use it step by step to show that the derivative of x² is 2x. You will practise the same algebra on a new function.
1 code6 build steps3 exercises3 questions - 3
The derivative as a slope: tangent lines and secant approximations Beyond Pass
You learn that the derivative f'(x) gives the slope of the tangent line at a point on a curve. You also see how the slope of a secant line through two nearby points approximates that tangent slope.
1 code3 build steps3 exercises4 questions - 4
Differentiating polynomials and finding tangent slopes Beyond Pass
You learn what the derivative tells you (the slope of the tangent line), and how a secant slope between two close points approximates it. Then you differentiate polynomials term by term with the power rule and constant multiple rule, and substitute an x-value to find the slope of a tangent line.
2 code3 formulas2 build steps3 exercises3 questions - 5
Reciprocal functions and negative exponents Beyond Pass
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. You then rewrite the answer back into fraction form and handle square and cube roots the same way.
3 exercises4 questions - 6
Differentiating radical functions with rational exponents Beyond Pass
You learn to turn a radical into a power with a fractional exponent, then apply the power rule with fraction arithmetic. You finish by rewriting the answer as a radical again.
3 exercises3 questions - 7
Simplifying before differentiating: distributing, expanding and dividing Beyond Pass
Before taking a derivative, you can often rewrite the function in a simpler polynomial form by distributing, expanding a squared binomial, or dividing each term by the denominator. Once the function is a sum of simple powers of x, the power rule finds the derivative term by term.
3 code4 build steps3 exercises3 questions - 8
Derivatives of trigonometric functions Beyond Pass
This lesson gives you the six basic derivatives of the trigonometric functions: sine, cosine, secant, cosecant, tangent and cotangent. You will learn a memory trick for the signs and practise using the results.
- 9
The product rule for two and three factors Beyond Pass
The product rule tells you how to differentiate a product of two functions: take the derivative of the first times the second, plus the first times the derivative of the second. It extends to three factors by differentiating one factor at a time while the others stay the same.
3 exercises3 questions - 10
The quotient rule for rational functions Beyond Pass
The quotient rule gives the derivative of a fraction f/g as (g·f' − f·g') / g². You learn to name the top and bottom, differentiate each, and assemble and simplify the result.
3 exercises3 questions
Final project
Build a Derivative Toolkit and Tangent-Line Checker
Make a small Python program that differentiates a set of functions by the rules from this course and checks each result numerically against secant slopes. It puts the power, product and quotient rules, and the trig derivatives, together with the idea that a derivative is a tangent slope.
Its 6 tasks open with Beyond Pass.
Free preview · Lesson 1 in full
Derivatives of constants and the power rule
You will learn that the derivative of a constant is zero, and how the power rule and the constant multiple rule let you differentiate monomials like x^n and c·x^n. You will practise on several examples and check each answer step by step.
Run and change the code, with real Python and JavaScript.
Draw and write next to the video.
The idea
A derivative tells you the slope of a function at any x value. Writing d/dx in front of a function means 'differentiate this with respect to x', and the result is written f'(x). For a constant such as f(x) = 8, the graph is a flat horizontal line, and a flat line has slope zero. So the derivative of any constant is zero: the derivative of 5 is 0, and the derivative of -7 is 0.
The power rule handles monomials, which are a variable raised to a constant power. For any power n, the derivative of x^n is n times x to the power n minus 1. You bring the exponent down to multiply in front, then subtract 1 from the exponent. For x^2, n = 2, so the derivative is 2x^(2-1) = 2x^1 = 2x. For x^3 you get 3x^2, for x^4 you get 4x^3, and for x^5 you get 5x^4.
When a number multiplies the variable, such as 4x^7, use the constant multiple rule: the derivative of c times f(x) is c times the derivative of f(x). In other words, you move the constant c outside, differentiate the rest, and then multiply. So the derivative of 4x^7 is 4 times (7x^6), which is 28x^6. The same pattern works for 8x^4, giving 8 times 4x^3 = 32x^3, and for 5x^6, giving 5 times 6x^5 = 30x^5.
Here is the full method in three quick steps. First, identify the constant multiplier c (if there is one) and the power n. Second, multiply c by n, which is the new coefficient. Third, lower the power by one, so the new exponent is n minus 1. For 9x^5: the coefficient becomes 9 times 5 = 45, and the exponent becomes 4, giving 45x^4. For 6x^7: 6 times 7 = 42, and the exponent becomes 6, giving 42x^6.
Key points
- The derivative of any constant is zero, because a constant function is a flat line with slope zero. 0:07
- Power rule: the derivative of x^n is n·x^(n-1). Multiply by the old exponent, then subtract 1 from it. 2:07
- Constant multiple rule: the derivative of c·f(x) is c times the derivative of f(x), so the constant just comes along for the ride. 4:00
- For 4x^7 the answer is 4 × 7x^6 = 28x^6: multiply the coefficient 4 by the exponent 7, then reduce the exponent by one. 5:11
- For 8x^4 the answer is 32x^3, and for 5x^6 the answer is 30x^5. 6:41
- For 9x^5 the answer is 45x^4, and for 6x^7 the answer is 42x^6. 7:51
Build along in the practice terminal
- Write down a function that is a constant, for example f(x) = 8, and state its derivative. You should get 0, because a flat line has slope zero.
- Take the function x^2, identify n = 2, and apply the power rule: bring the 2 down and reduce the exponent by 1. You should get 2x.
- Differentiate x^3, x^4 and x^5 with the power rule. You should get 3x^2, 4x^3 and 5x^4.
- Differentiate 4x^7: move the 4 out front, differentiate x^7 to get 7x^6, then multiply 4 times 7. You should get 28x^6.
Practice
Find the derivative of 3x^5 and evaluate it at x = 2. Give the numeric value.
Hint
Keep the 3 out front, use the power rule on x^5, then put x = 2 into your result.
Solution
- Apply the constant multiple rule: d/dx(3x^5) = 3 · d/dx(x^5).
- Power rule: d/dx(x^5) = 5x^4, so the derivative is 3 · 5x^4 = 15x^4.
- Substitute x = 2: 15 · 2^4 = 15 · 16 = 240.
240
Write a Python function deriv_monomial(c, n, x) that returns the derivative of c·x^n evaluated at x, using the rule c·n·x^(n-1). Test it with c = 4, n = 7, x = 1.
def deriv_monomial(c, n, x):
# return the derivative of c*x**n at x
pass
print(deriv_monomial(4, 7, 1))Solution
def deriv_monomial(c, n, x):
return c * n * x ** (n - 1)
print(deriv_monomial(4, 7, 1))28
Explain in your own words why the derivative of the constant 8 is zero, using the idea of slope.
Solution
- The graph of f(x) = 8 is a horizontal straight line at height 8.
- The derivative gives the slope at each x, and the slope of a horizontal line is zero everywhere.
- So the derivative of 8 is 0.
A constant function is a horizontal line, and a horizontal line has slope 0 at every point, so its derivative is 0.
On your scratchpad
Sketch a flat horizontal line at height 8 on your scratchpad, then draw a small tangent line at two points on it and write the slope next to each one.
Quiz
What is the derivative of the constant -7?
- -7
- 0
- 1
- -7x
Show the answer
B. 0 · A constant function is a flat horizontal line with slope zero, so its derivative is 0 at every x.
Using the power rule, what is the derivative of x^4?
- x^3
- 4x^3
- 4x^4
- 3x^4
Show the answer
B. 4x^3 · Here n = 4, so bring the 4 down and reduce the exponent to 4 - 1 = 3, giving 4x^3.
What is the derivative of 5x^6?
- 30x^5
- 5x^5
- 6x^5
- 30x^6
Show the answer
A. 30x^5 · Keep the constant 5 out front: 5 times (6x^5) = 30x^5.
What is the derivative of 9x^5?
- 45x^4
- 9x^4
- 45x^5
- 5x^4
Show the answer
A. 45x^4 · The constant 9 times the power rule result 5x^4 gives 45x^4.
Lessons 2–10 and the final project
They open with Beyond Pass: every lesson of every course, in full. One payment: $14.99 for 90 days or $29 for a year. No subscription.
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