Find the derivative of y with respect to x for the given functions and evaluate the limit.

Here are the solutions to the problems: Problem 1: إذا كانت ص = س^2 + 3س - 11، فإن (د ص)/(د س) To find the derivative (dy)/(dx), we apply the power rule for differentiation. Step 1: Differentiate each term with respect to x. (d)/(dx)(x^2) = 2x (d)/(dx)(3x) = 3 (d)/(dx)(-11) = 0 Step 2: Combine the derivatives. (dy)/(dx) = 2x + 3 + 0 (dy)/(dx) = 2x + 3 The derivative is 2x + 3 Problem 2: إذا كانت ص = (س - 3)^4، فإن (د ص)/(د س) To find the derivative (dy)/(dx), we use the chain rule. Step 1: Apply the power rule to the outer function and multiply by the derivative of the inner function. (dy)/(dx) = 4(x - 3)^4-1 · (d)/(dx)(x - 3) Step 2: Differentiate the inner function. (d)/(dx)(x - 3) = 1 - 0 = 1 Step 3: Substitute the derivative of the inner function back into the expression. (dy)/(dx) = 4(x - 3)^3 · 1 (dy)/(dx) = 4(x - 3)^3 The derivative is 4(x - 3)^3 Problem 3: نها س 4 (س - 4)/(س^2 - 16) To evaluate the limit, we first try direct substitution, which gives (0)/(0), an indeterminate form. We need to simplify the expression. Step 1: Factor the denominator using the difference of squares formula (a^2 - b^2 = (a - b)(a + b)). x^2 - 16 = (x - 4)(x + 4) Step 2: Substitute the factored denominator back into the limit expression. _x 4 (x - 4)/((x - 4)(x + 4)) Step 3: Cancel out the common factor (x - 4) (since x ≠ 4 as x approaches 4). _x 4 (1)/(x + 4) Step 4: Now, substitute x = 4 into the simplified expression. (1)/(4 + 4) = (1)/(8) The limit is (1)/(8) Problem 4: إذا كانت د(س) = 3س - 11، فإن د'(س) To find the derivative f'(x), we apply the power rule for differentiation. Step 1: Differentiate each term with respect to x. (d)/(dx)(3x) = 3 (d)/(dx)(-11) = 0 Step 2: Combine the derivatives. f'(x) = 3 - 0 f'(x) = 3 The derivative is 3 Problem 5: إذا كانت د(س) = 10، فإن د'(س) To find the derivative f'(x), we differentiate the constant function. Step 1: The derivative of any constant is 0. f'(x) = 0 The derivative is 0 Problem 6: إذا كانت د(س) = (16)/(س)، فإن د'(س) To find the derivative f'(x), we first rewrite the function using negative exponents. Step 1: Rewrite f(x) as 16x^-1. f(x) = 16x^-1 Step 2: Apply the power rule for differentiation. f'(x) = 16 · (-1)x^-1-1 f'(x) = -16x^-2 Step 3: Rewrite the expression with a positive exponent. f'(x) = -(16)/(x^2) The derivative is -(16)/(x^2) Problem 7: إذا كانت د(س) = س^3 + 9س + 1، فإن د'(س) To find the derivative f'(x), we apply the power rule for differentiation to each term. Step 1: Differentiate each term with respect to x. (d)/(dx)(x^3) = 3x^2 (d)/(dx)(9x) = 9 (d)/(dx)(1) = 0 Step 2: Combine the derivatives. f'(x) = 3x^2 + 9 + 0 f'(x) = 3x^2 + 9 The derivative is 3x^2 + 9 Problem 8: إذا كانت $د(س) = (س - 3)(س