Turunan Fungsi Aljabar Khusus

Explanation
The given function is a product of two functions, (6x - 3)^3 and (2x - 1). To find the derivative of this function, we can use the product rule. The derivative of the first function is 3(6x - 3)^2 * 6, and the derivative of the second function is 2. Multiplying these two derivatives together, we get f'(x) = 36(6x - 3)^2 * 2. Plugging in x = 1, we get f'(1) = 36(6 - 3)^2 * 2 = 36(3)^2 * 2 = 36 * 9 * 2 = 648. Therefore, the correct answer is E. 216.

Explanation
The given function is a product of two functions: (x^3 - 2x) and (x^2 + 3). To find the derivative of this function, we can use the product rule. The derivative of the first function (x^3 - 2x) is 3x^2 - 2, and the derivative of the second function (x^2 + 3) is 2x. Applying the product rule, we get (x^2 + 3)(3x^2 - 2) + (x^3 - 2x)(2x). Simplifying this expression gives us 5x^4 + 3x^2 - 6, which matches the given answer e.

Explanation
The derivative of a function is the rate at which the function is changing at a particular point. To find the derivative of f(x) = 2x^2 - 5x + 3, we can apply the power rule for derivatives. The power rule states that the derivative of x^n is equal to n*x^(n-1). Applying this rule, we get f'(x) = 4x - 5. To find f'(2), we substitute x = 2 into the derivative function, giving us f'(2) = 4(2) - 5 = 8 - 5 = 3. Therefore, the correct answer is C. 3.

Explanation
The given function is a product of two functions, (2x - 2) and (1 - x^2). To find the derivative of this function, we can use the product rule. The product rule states that the derivative of a product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function. Applying this rule to the given function, we get f'(x) = (2(1 - x^2)) + (2x(2x - 2)). Simplifying this expression gives us f'(x) = 2(1 - x^2) + 2x(2x - 2), which matches option B.

Explanation
The first derivative of the function f(x) is f'(x). If f'(k) = 11, then the possible value of (k+1) can be found by substituting k into the first derivative and solving for (k+1). In this case, if f'(k) = 11, then substituting -1 into the first derivative would give us 11. Therefore, the possible value of (k+1) is -1.

Explanation
The question asks for the value of f'(3), which represents the derivative of the function f(x) at x=3. To find the derivative of f(x), we can use the power rule of differentiation. Taking the derivative of x^2 gives us 2x, and the derivative of -3x is -3. Therefore, the derivative of f(x) is f'(x) = 2x - 3. To find f'(3), we substitute x=3 into the derivative equation, giving us f'(3) = 2(3) - 3 = 6 - 3 = 3. Therefore, the correct answer is B. 3.

Explanation
The given function f(x) = (2x-1)^2 is a quadratic function in the form of (ax+b)^2, where a = 2 and b = -1. To find the value of x when f'(x) = 0, we need to find the derivative of f(x) and set it equal to 0. Taking the derivative of f(x) using the power rule, we get f'(x) = 2(2x-1)(2) = 4(2x-1). Setting this equal to 0 and solving for x, we get 2x-1 = 0, which gives x = 1. Therefore, the correct answer is E. 1.

Explanation
The given function is f(x) = (6x - 3)^3 (2x - 1). To find the first derivative of this function, we can use the product rule and the chain rule. After differentiating, we get f'(x) = 3(6x - 3)^2 (2x - 1) + (6x - 3)^3 (2). To find f'(1), we substitute x = 1 into the derivative equation. Simplifying, we get f'(1) = 3(6 - 3)^2 (2 - 1) + (6 - 3)^3 (2) = 3(3)^2 (1) + (3)^3 (2) = 3(9) + 27(2) = 27 + 54 = 81. Therefore, the correct answer is C. 54.

Explanation
The given expression is a product of two functions, y = x^2 and (x^2 - 1). To find the derivative of this expression, we can use the product rule. The derivative of the first function, x^2, is 2x. The derivative of the second function, (x^2 - 1), is also 2x. Applying the product rule, we get the derivative of the expression as (x^2)(2x) + (2x)(x^2 - 1) = 2x^3 + 2x^3 - 2x = 4x^3 - 2x. Therefore, the correct answer is A. 4x^3 - 2x.

Explanation
The question states that F'(x) = 19. The derivative of F(x) is given by F'(x) = 8x + 3. Setting this equal to 19, we can solve for x: 8x + 3 = 19. Subtracting 3 from both sides gives 8x = 16, and dividing by 8 gives x = 2. Therefore, the value of x is 2, which corresponds to answer choice B.