Mathematics Grade 12 Worksheet on Unit Two

To differentiate y = (3x^2 + 2)^5, use the chain rule. The derivative is 5(3x^2 + 2)^4 * (6x) = 30x(3x^2 + 2)^4. Thus, the correct answer is 30x(3x^2 + 2)^4.

To find the derivative of y = sin(2x^3), use the chain rule. The derivative is cos(2x^3) multiplied by the derivative of 2x^3, which is 6x^2. Thus, the final answer is 6x^2 cos(2x^3).

To find dy/dx for y = e^(5x^2 + 3x), use the chain rule. The derivative of e^u is e^u * du/dx. Here, u = 5x^2 + 3x, so du/dx = 10x + 3. Thus, dy/dx = e^(5x^2 + 3x) * (10x + 3), which is the correct choice.

To differentiate y = ln(4x^2 + 1), use the chain rule. The derivative is (1/(4x^2 + 1)) * (8x) = 8x / (4x^2 + 1). Thus, the correct answer is 8x / (4x^2 + 1).

To find the derivative of y = (x^2 + 1)(3x - 4), use the product rule: y' = (x^2 + 1)(3) + (3x - 4)(2x). Simplifying gives y' = 9x^2 - 8x + 3, which matches the correct answer.

To find dy/dx for y = cos(5x^2 + 2x), use the chain rule. The derivative of cos(u) is -sin(u) * du/dx. Here, u = 5x^2 + 2x, so du/dx = 10x + 2. Thus, dy/dx = -sin(5x^2 + 2x) * (10x + 2), which matches the correct choice.

To differentiate y = (2x + 3)^4, use the chain rule. The derivative is 4(2x + 3)^3 multiplied by the derivative of (2x + 3), which is 2. Thus, the final result is 8(2x + 3)^3, making it the correct choice.

To find the derivative of y = tan(3x^2), use the chain rule. The derivative is sec^2(3x^2) multiplied by the derivative of 3x^2, which is 6x. Thus, the correct answer is 6x sec^2(3x^2).

To find dy/dx for y = (x^3 + 1)/(x - 1), use the quotient rule: dy/dx = (f'g - fg')/g^2. Here, f = x^3 + 1 and g = x - 1. After applying the rule, we get (2x^3 - 3x^2 - 1)/(x - 1)^2, which matches the correct answer.

To differentiate y = sqrt(5x^2 + 4x), use the chain rule. The derivative is (1/2)(5x^2 + 4x)^(-1/2) * (10x + 4). Simplifying gives (10x + 4) / (2sqrt(5x^2 + 4x)), which is the correct choice.