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Derivatives, integrals, trig identities and more — for advanced math learners!
1. What is sin(30°)?
💡 sin(30°) = 1/2. This is a fundamental trigonometric value.
2. What is the second derivative of x⁴?
💡 First derivative: 4x³. Second derivative: 12x².
3. What is the integral of 2x dx?
💡 ∫2x dx = x² + C, where C is the constant of integration.
4. What is the area under the curve y = x from x=0 to x=4?
💡 ∫₀⁴ x dx = [x²/2]₀⁴ = 16/2 - 0 = 8.
5. What is the derivative of x³?
💡 d/dx(xⁿ) = nxⁿ⁻¹. So d/dx(x³) = 3x².
6. What is cos(60°)?
💡 cos(60°) = 1/2. This is a standard trigonometric value.
7. What is cos(0°)?
💡 cos(0°) = 1. The cosine of zero degrees equals one.
8. What is the derivative of eˣ?
💡 The derivative of eˣ is eˣ itself — this is the unique property of the natural exponential function.
9. What does the derivative of a function represent geometrically?
💡 The derivative at a point gives the slope of the tangent line to the curve at that point.
10. What is the limit of (sin x)/x as x approaches 0?
💡 lim(x→0) sin(x)/x = 1. This is a fundamental limit in calculus.
11. What is ∫x² dx?
💡 ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. So ∫x² dx = x³/3 + C.
12. What is the derivative of ln(x)?
💡 d/dx(ln x) = 1/x. The derivative of the natural logarithm.
13. What is tan(45°)?
💡 tan(45°) = sin(45°)/cos(45°) = 1. Both sin and cos equal √2/2 at 45°.
14. What is the derivative of sin(x)?
💡 d/dx(sin x) = cos x. This is a fundamental calculus rule.
15. What is sin(90°)?
💡 sin(90°) = 1. This is one of the fundamental trigonometric values.
16. What is the integral of cos(x) dx?
💡 ∫cos(x) dx = sin(x) + C.
17. Which rule is used to differentiate a product of two functions?
💡 The product rule states: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x).
18. What is the value of sin²(x) + cos²(x)?
💡 This is the Pythagorean identity: sin²(x) + cos²(x) = 1, always true for any x.
19. In a right triangle, if the opposite side is 3 and hypotenuse is 5, what is sin(θ)?
💡 sin(θ) = opposite/hypotenuse = 3/5.
20. What is the derivative of x⁵ - 3x³ + 2x?
💡 d/dx(x⁵) = 5x⁴, d/dx(-3x³) = -9x², d/dx(2x) = 2. Total: 5x⁴ - 9x² + 2.