Find the equation of the tangent to the curve y = 5 - 2x - x2 at the point (1, 2). Use the quotient rule to differentiate y = 3x/(x2+1).

Step 1: Find the derivative of the curve. Given y = 5 - 2x - x^2. Differentiating with respect to x: (dy)/(dx) = (d)/(dx)(5) - (d)/(dx)(2x) - (d)/(dx)(x^2) (dy)/(dx) = 0 - 2 - 2x (dy)/(dx) = -2 - 2x Step 2: Calculate the slope of the tangent at the point (1, 2). Substitute x=1 into the derivative: m = -2 - 2(1) m = -2 - 2 m = -4 Step 3: Use the point-slope form to find the equation of the tangent. The point is (x_1, y_1) = (1, 2) and the slope is m = -4. The equation of a line is y - y_1 = m(x - x_1). y - 2 = -4(x - 1) y - 2 = -4x + 4 y = -4x + 4 + 2 y = -4x + 6 The equation of the tangent is y = -4x + 6. Question 4: Use the quotient rule to differentiate y = (3x+x^4)/(2x^2+1). Step 1: Identify u and v and find their derivatives. Let u = 3x+x^4. Then u' = (d)/(dx)(3x+x^4) = 3 + 4x^3. Let v = 2x^2+1. Then v' = (d)/(dx)(2x^2+1) = 4x. Step 2: Apply the quotient rule formula (dy)/(dx) = (v u' - u v')/(v^2). (dy)/(dx) = ((2x^2+1)(3+4x^3) - (3x+x^4)(4x))/((2x^2+1)^2) Step 3: Expand and simplify the numerator. Numerator: (2x^2+1)(3+4x^3) = 6x^2 + 8x^5 + 3 + 4x^3 = 8x^5 + 4x^3 + 6x^2 + 3 (3x+x^4)(4x) = 12x^2 + 4x^5 Subtracting the second from the first: (8x^5 + 4x^3 + 6x^2 + 3) - (12x^2 + 4x^5) = 8x^5 + 4x^3 + 6x^2 + 3 - 12x^2 - 4x^5 = 4x^5 + 4x^3 - 6x^2 + 3 Step 4: Write the final derivative. (dy)/(dx) = (4x^5 + 4x^3 - 6x^2 + 3)/((2x^2+1)^2) Question 5: Find f'(x). i. f(x) = 5 (3x^2) Step 1: Apply the chain rule. Let u = 3x^2. Then (du)/(dx) = 6x. The derivative of 5 (u) with respect to u is 5 (u). Step 2: Multiply the derivatives. f'(x) = 5 (u) · (du)/(dx) f'(x) = 5 (3x^2) · (6x) f'(x) = 30x (3x^2) ii. f(x) = 4 (sqrt(x)) Step 1: Apply the chain rule. Let u = sqrt(x) = x^1/2. Then (du)/(dx) = (1)/(2)x^-1/2 = (1)/(2sqrt(x)). The derivative of 4 (u) with respect to u is 4 ^2(u). Step 2: Multiply the derivatives. f'(x) = 4 ^2(u) · (du)/(dx) f'(x) = 4 ^2(sqrt(x)) · ((1)/(2sqrt(x))) f'(x) = 2 ^2(sqrt(x))sqrt(x) Send me the next one 📸