This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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1. Find the derivative of y = 3x^4 - 5x^2 + 7x - 2 with respect to x. Step 1: Apply the power rule (d)/(dx)(ax^n) = anx^n-1 and the constant rule (d)/(dx)(c) = 0 to each term. (dy)/(dx) = (d)/(dx)(3x^4) - (d)/(dx)(5x^2) + (d)/(dx)(7x) - (d)/(dx)(2) (dy)/(dx) = 3(4)x^4-1 - 5(2)x^2-1 + 7(1)x^1-1 - 0 Step 2: Simplify the expression. (dy)/(dx) = 12x^3 - 10x^1 + 7x^0 (dy)/(dx) = 12x^3 - 10x + 7 The correct option is (A). A Send me the next one 📸 2. If y = (2x + 3)^5, find dy/dx. Step 1: Apply the chain rule (d)/(dx)(u^n) = nu^n-1 (du)/(dx), where u = 2x + 3 and n = 5. (dy)/(dx) = 5(2x + 3)^5-1 · (d)/(dx)(2x + 3) Step 2: Differentiate the inner function (d)/(dx)(2x + 3). (d)/(dx)(2x + 3) = 2(1)x^1-1 + 0 = 2 Step 3: Substitute the derivative of the inner function back into the expression. (dy)/(dx) = 5(2x + 3)^4 · 2 (dy)/(dx) = 10(2x + 3)^4 The correct option is (B). B What's next? 3. Differentiate y = x^3 (x) with respect to x. Step 1: Apply the product rule (d)/(dx)(uv) = u'v + uv', where u = x^3 and v = (x). First, find u' and v'. u = x^3 u' = (d)/(dx)(x^3) = 3x^2 v = (x) v' = (d)/(dx)((x)) = (x) Step 2: Substitute u, u', v, v' into the product rule formula. (dy)/(dx) = (3x^2)((x)) + (x^3)((x)) (dy)/(dx) = 3x^2 (x) + x^3 (x) The correct option is (D). D Got more? Send 'em. 4. Find the derivative of y = (3x - 1) / (2x + 5). Step 1: Apply the quotient rule (d)/(dx)((u)/(v)) = (u'v - uv')/(v^2), where u = 3x - 1 and v = 2x + 5. First, find u' and v'. u = 3x - 1 u' = (d)/(dx)(3x - 1) = 3 v = 2x + 5 v' = (d)/(dx)(2x + 5) = 2 Step 2: Substitute u, u', v, v' into the quotient rule formula. (dy)/(dx) = ((3)(2x + 5) - (3x - 1)(2))/((2x + 5)^2) Step 3: Expand and simplify the numerator. (dy)/(dx) = (6x + 15 - (6x - 2))/((2x + 5)^2) (dy)/(dx) = (6x + 15 - 6x + 2)/((2x + 5)^2) (dy)/(dx) = (17)/((2x + 5)^2) The correct option is (A). A Drop the next question. 5. Evaluate dy/dx at x = 2 for the curve y = 2x^3 - 4x + 5. Step 1: Find the derivative (dy)/(dx) using the power rule. (dy)/(dx) = (d)/(dx)(2x^3) - (d)/(dx)(4x) + (d)/(dx)(5) (dy)/(dx) = 2(3)x^3-1 - 4(1)x^1-1 + 0 (dy)/(dx) = 6x^2 - 4 Step 2: Substitute x = 2 into the derivative. (dy)/(dx)|_x=2 = 6(2)^2 - 4 (dy)/(dx)|_x=2 = 6(4) - 4 (dy)/(dx)|_x=2 = 24 - 4 (dy)/(dx)|_x=2 = 20 The correct option is (A). A Send me the next one 📸 6. Find the gradient of the curve y = x^2 - 3x + 2 at the point where x = 4. Step 1: Find the derivative (dy)/(dx) (which represents the gradient) using the power rule. (dy)/(dx) = (d)/(dx)(x^2) - (d)/(dx)(3x) + (d)/(dx)(2) (dy)/(dx) = 2x^2-1 - 3(1)x^1-1 + 0 (dy)/(dx) = 2x - 3 Step 2: Substitute x = 4 into the derivative to find the gradient at that point. Gradient = 2(4) - 3 Gradient = 8 - 3 Gradient = 5 The correct option is (A). A What's next? 7. Find the stationary points (turning points) of the curve y = x^3 - 3x + 4. Step 1: Find the derivative (dy)/(dx). (dy)/(dx) = (d)/(dx)(x^3) - (d)/(dx)(3x) + (d)/(dx)(4) (dy)/(dx) = 3x^2 - 3 Step 2: Set the derivative to zero to find the x-coordinates of the stationary points. 3x^2 - 3 = 0 3x^2 = 3 x^2 = 1 x = ±sqrt(1) x = 1 or x = -1 The correct option is (A). A Got more? Send 'em. 8. Determine the minimum value of the function y = x^2 - 6x + 10. Step 1: Find the derivative (dy)/(dx). (dy)/(dx) = (d)/(dx)(x^2) - (d)/(dx)(6x) + (d)/(dx)(10) (dy)/(dx) = 2x - 6 Step 2: Set the derivative to zero to find the x-coordinate of the stationary point. 2x - 6 = 0 2x = 6 x = 3 Step 3: Substitute x = 3 back into the original function to find the minimum value. y = (3)^2 - 6(3) + 10 y = 9 - 18 + 10 y = -9 + 10 y = 1 The correct option is (A). A Drop the next question. 9. If the displacement of a particle is given by s = 2t^3 - 9t^2 + 12t + 5, find its acceleration when t = 2 seconds. Step 1: Find the velocity v by differentiating the displacement s with respect to time t. v = (ds)/(dt) = (d)/(dt)(2t^3 - 9t^2 + 12t + 5) v = 2(3)t^2 - 9(2)t^1 + 12(1)t^0 + 0 v = 6t^2 - 18t + 12 Step 2: Find the acceleration a by differentiating the velocity v with respect to time t. a = (dv)/(dt) = (d)/(dt)(6t^2 - 18t + 12) a = 6(2)t^1 - 18(1)t^0 + 0 a = 12t - 18 Step 3: Substitute t = 2 seconds into the acceleration equation. a = 12(2) - 18 a = 24 - 18 a = 6 m/s^2 The correct option is (A). A Send me the next one 📸 10. Differentiate y = (4x^3 - 2x) with respect to x. Step 1: Apply the chain rule (d)/(dx)((u)) = -(u) (du)/(dx), where u = 4x^3 - 2x. First, find (du)/(dx). u = 4x^3 - 2x (du)/(dx) = (d)/(dx)(4x^3 - 2x) = 4(3)x^2 - 2(1) = 12x^2 - 2 Step 2: Substitute u and (du)/(dx) into the chain rule formula. (dy)/(dx) = -(4x^3 - 2x) · (12x^2 - 2) (dy)/(dx) = -(12x^2 - 2) (4x^3 - 2x) The correct option is (A). A What's next? 11. Find dy/dx if y = e^5x^2. Step 1: Apply the chain rule (d)/(dx)(e^u) = e^u (du)/(dx), where u = 5x^2. First, find (du)/(dx). u = 5x^2 (du)/(dx) = (d)/(dx)(5x^2) = 5(2)x^2-1 = 10x Step 2: Substitute u and (du)/(dx) into the chain rule formula. (dy)/(dx) = e^5x^2 · (10x) (dy)/(dx) = 10x e^5x^2 The correct option is (A). A Got more? Send 'em. 12. Differentiate y = (3x^2 + 4) with respect to x. Step 1: Apply the chain rule (d)/(dx)((u)) = (1)/(u) (du)/(dx), where u = 3x^2 + 4. First, find (du)/(dx). u = 3x^2 + 4 (du)/(dx) = (d)/(dx)(3x^2 + 4) = 3(2)x^2-1 + 0 = 6x Step 2: Substitute u and (du)/(dx) into the chain rule formula. (dy)/(dx) = (1)/(3x^2 + 4) · (6x) (dy)/(dx) = (6x)/(3x^2 + 4) The correct option is (A). A Drop the next question. 13. Given the implicit equation x^2 + y^2 = 25, find dy/dx. Step 1: Differentiate both sides of the equation with respect to x. Remember that y is a function of x, so use the chain rule for y^2. (d)/(dx)(x^2) + (d)/(dx)(y^2) = (d)/(dx)(25) 2x + 2y (dy)/(dx) = 0 Step 2: Isolate (dy)/(dx). 2y (dy)/(dx) = -2x (dy)/(dx) = (-2x)/(2y) (dy)/(dx) = -(x)/(y) The correct option is (A). A Send me the next one 📸 14. Find the rate of change of the area of a circle with respect to its radius r when r = 7 cm. Step 1: Write down the formula for the area of a circle. A = r^2 Step 2: Differentiate the area A with respect to the radius r. (dA)/(dr) = (d)/(dr)( r^2) (dA)/(dr) = (2r) (dA)/(dr) = 2 r Step 3: Substitute r = 7 cm into the derivative. (dA)/(dr)|_r=7 = 2 (7) (dA)/(dr)|_r=7 = 14 cm^2/cm The correct option is (A). A What's next? 15. If y = x (x), find dy/dx. Step 1: Apply the product rule (d)/(dx)(uv) = u'v + uv', where u = x and v = (x). First, find u' and v'. u = x u' = (d)/(dx)(x) = 1 v = (x) v' = (d)/(dx)((x)) = ^2(x) Step 2: Substitute u, u', v, v' into the product rule formula. (dy)/(dx) = (1)((x)) + (x)(^2(x)) (dy)/(dx) = (x) + x ^2(x) The correct option is (A). A Got more? Send 'em.