01 Key Concepts
Power Rule
d/dx[x^n] = n*x^(n-1).
Constant Multiple Rule
d/dx[c*f(x)] = c*f'(x) — constants pass through the derivative unchanged.
Sum/Difference Rule
d/dx[f(x) +/- g(x)] = f'(x) +/- g'(x) — differentiate term by term.
Product Rule
d/dx[f(x)*g(x)] = f'(x)*g(x) + f(x)*g'(x).
Quotient Rule
d/dx[f(x)/g(x)] = [f'(x)*g(x) - f(x)*g'(x)] / [g(x)]^2.
Derivatives of Common Functions
d/dx[sin x] = cos x, d/dx[cos x] = -sin x, d/dx[e^x] = e^x, d/dx[ln x] = 1/x.
02 Key Formulas
- d/dx[c*f(x)] = c*f'(x)
- d/dx[f+g] = f' + g'
- d/dx[f*g] = f'g + fg'
- d/dx[f/g] = (f'g - fg') / g^2
03 Solved Examples
Example 1 Differentiate f(x) = 5x^3 + 2x^2 - 7x + 4.
- Apply the power rule to each term: d/dx[5x^3] = 15x^2.
- d/dx[2x^2] = 4x.
- d/dx[-7x] = -7.
- d/dx[4] = 0.
- Combine all terms.
Answer: f'(x) = 15x^2 + 4x - 7
Example 2 Differentiate f(x) = x^2 * sin(x) using the product rule.
- Let u = x^2 (u' = 2x) and v = sin(x) (v' = cos(x)).
- Apply product rule: u'v + uv' = 2x*sin(x) + x^2*cos(x).
Answer: f'(x) = 2x*sin(x) + x^2*cos(x)
Example 3 Differentiate f(x) = x / (x+1) using the quotient rule.
- Let u = x (u'=1) and v = x+1 (v'=1).
- Apply quotient rule: (u'v - uv')/v^2 = [1*(x+1) - x*1] / (x+1)^2.
- Simplify the numerator: (x+1-x) = 1.
Answer: f'(x) = 1 / (x+1)^2
04 Practice Questions
1Differentiate f(x) = 4x^3 - 2x + 1.
12x^2 - 2
2Differentiate f(x) = x^2 * cos(x) using the product rule.
2x*cos(x) - x^2*sin(x)
3Differentiate f(x) = (x+2)/(x-1) using the quotient rule.
-3 / (x-1)^2
4Differentiate f(x) = 6x^5.
30x^4
5Differentiate f(x) = e^x + ln(x).
e^x + 1/x
📄 Rules of Differentiation — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.