1
The derivative of f(x) = 4 x^3 - 5 x^2 + 7 x - 9 with respect to x is:
التفسير والحل والبرهان النموذجي: Power rule: d/dx [a*x^n] = n*a*x^(n-1). d/dx [4x^3 - 5x^2 + 7x - 9] = 12x^2 - 10x + 7.
DABOUR HUB / QUIZ
الاختبار التفاعلي والتقييم: Lesson 1 - Rate of Change and Differentiation Rules
2
The slope of the tangent line to the curve y = x^2 - 6 x + 5 at the point (4, -3) equals:
التفسير والحل والبرهان النموذجي: dy/dx = 2x - 6. At x = 4: m = 2(4) - 6 = 8 - 6 = 2.
3
If y = (2x + 1) * (3x - 4), then dy/dx evaluated at x = 1 equals:
التفسير والحل والبرهان النموذجي: y = 6x^2 - 5x - 4 => dy/dx = 12x - 5. At x = 1: 12(1) - 5 = 7... wait: Product rule: (2)(3x-4) + (2x+1)(3) = 2(3-4) + (3)(3) = -2 + 9 = 7. Option A in list was 1, option D is 7.
4
Using the Quotient Rule, the derivative of y = (x + 3) / (x - 2) for x != 2 is:
التفسير والحل والبرهان النموذجي: Quotient rule: [ (1)*(x-2) - (x+3)*(1) ] / (x-2)^2 = [ x - 2 - x - 3 ] / (x-2)^2 = -5 / (x-2)^2.
5
According to the Chain Rule, if y = (3 x^2 - 5)^4, then dy/dx is:
التفسير والحل والبرهان النموذجي: Chain rule: dy/dx = 4 (3x^2 - 5)^3 * d/dx(3x^2 - 5) = 4 (3x^2 - 5)^3 * (6x) = 24 x (3x^2 - 5)^3.
6
The derivative of y = sin(3x) with respect to x is:
التفسير والحل والبرهان النموذجي: d/dx [sin(kx)] = k cos(kx) => d/dx [sin(3x)] = 3 cos(3x).
7
If a particle moves along a straight line such that its position s(t) = t^3 - 6 t^2 + 9 t meters, the particle comes to an instantaneous rest (velocity v(t) = 0) at times:
التفسير والحل والبرهان النموذجي: v(t) = s'(t) = 3t^2 - 12t + 9 = 0 => 3(t^2 - 4t + 3) = 0 => (t - 1)(t - 3) = 0 => t = 1 s and t = 3 s.
8
The equation of the tangent line to the curve y = x^3 at the point (1, 1) is:
التفسير والحل والبرهان النموذجي: dy/dx = 3x^2. At x = 1, m = 3. Line equation: y - 1 = 3(x - 1) => y - 1 = 3x - 3 => y = 3x - 2.
9
The second derivative d^2y/dx^2 of the function y = x^4 - 2 x^3 + 5 is:
التفسير والحل والبرهان النموذجي: y' = 4x^3 - 6x^2 => y'' = 12x^2 - 12x.
10
If f(x) = |x - 3|, the derivative f'(3) at x = 3:
التفسير والحل والبرهان النموذجي: Corner/cusp at x = 3: left derivative is -1 and right derivative is +1, so the two-sided limit does not exist.
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