Differentiate with respect to x. y = (cosx)2

here are the solutions to your math problems. 11. Differentiate with respect to x. y = ( x)^2x Step 1: Take the natural logarithm of both sides. y = (( x)^2x) y = 2x ( x) Step 2: Differentiate both sides with respect to x using implicit differentiation and the product rule. (1)/(y) (dy)/(dx) = 2 ( x) + 2x ( (1)/( x) · (- x) ) (1)/(y) (dy)/(dx) = 2 ( x) - 2x ( x)/( x) (1)/(y) (dy)/(dx) = 2 ( x) - 2x x Step 3: Solve for (dy)/(dx) by multiplying by y. (dy)/(dx) = y (2 ( x) - 2x x) Substitute y = ( x)^2x back into the equation. (dy)/(dx) = ( x)^2x (2 ( x) - 2x x) (dy)/(dx) = 2 ( x)^2x (( x) - x x) The derivative is: (dy)/(dx) = 2 ( x)^2x (( x) - x x) 12. A curve passes through the point (1,7) and the gradient of the curve at the point (x,y) is given by (5-2x). Find the equation of the curve. Step 1: Set up the differential equation. The gradient is given by (dy)/(dx). (dy)/(dx) = 5 - 2x Step 2: Integrate both sides with respect to x to find the equation of the curve. y = (5 - 2x) \, dx y = 5x - 2 (x^2)/(2) + C y = 5x - x^2 + C Step 3: Use the given point (1,7) to find the constant of integration C. Substitute x=1 and y=7 into the equation. 7 = 5(1) - (1)^2 + C 7 = 5 - 1 + C 7 = 4 + C C = 7 - 4 C = 3 Step 4: Write the final equation of the curve. y = 5x - x^2 + 3 13. Solve the equation z^2 = 5 - 12i. Step 1: Let z = x + iy, where x and y are real numbers. (x + iy)^2 = 5 - 12i x^2 + 2ixy + (iy)^2 = 5 - 12i x^2 + 2ixy - y^2 = 5 - 12i (x^2 - y^2) + i(2xy) = 5 - 12i Step 2: Equate the real and imaginary parts. x^2 - y^2 = 5 (1) 2xy = -12 (2) Step 3: From equation (2), express y in terms of x. y = (-12)/(2x) = (-6)/(x) Step 4: Substitute y into equation (1). x^2 - ((-6)/(x))^2 = 5 x^2 - (36)/(x^2) = 5 Multiply by x^2 to clear the denominator. x^4 - 36 = 5x^2 Rearrange into a quadratic form for x^2. x^4 - 5x^2 - 36 = 0 Step 5: Let u = x^2. Solve the quadratic equation u^2 - 5u - 36 = 0. Using the quadratic formula or factoring: (u - 9)(u + 4) = 0 So, u = 9 or u = -4. Since u = x^2 and x is a real number, x^2 must be non-negative. Therefore, x^2 = 9. x = ± 3 Step 6: Find the corresponding values for y. If x = 3, then y = (-6)/(3) = -2. So, z_1 = 3 - 2i. If x = -3, then y = (-6)/(-3) = 2. So, z_2 = -3 + 2i. The solutions are: z = 3 - 2i or z = -3 + 2i 14. Evaluate _0^2 (1 - |x - 1|)^3 dx. Step 1: Split the integral based on the definition of the absolute value. The expression |x-1| changes definition at x=1. For 0 x < 1: |x-1| = -(x-1) = 1-x. For 1 x 2: |x-1| = x-1. Step 2: Rewrite the integral as a sum of two integrals. _0^2 (1 - |x - 1|)^3 dx = _0^1 (1 - (1-x))^3 dx + _1^2 (1 - (x-1))^3 dx = _0^1 (1 - 1 + x)^3 dx + _1^2 (1 - x + 1)^3 dx = _0^1 x^3 dx + _1^2 (2 - x)^3 dx Step 3: Evaluate the first integral. _0^1 x^3 dx = [ (x^4)/(4) ]_0^1 = (1^4)/(4) - (0^4)/(4) = (1)/(4) Step 4: Evaluate the second integral. Let u = 2-x, so du = -dx. When x=1, u=2-1=1. When x=2, u=2-2=0. _1^2 (2 - x)^3 dx = _1^0 u^3 (-du) = - _1^0 u^3 du = _0^1 u^3 du = [ (u^4)/(4) ]_0^1 = (1^4)/(4) - (0^4)/(4) = (1)/(4) Step 5: Add the results of the two integrals. _0^2 (1 - |x - 1|)^3 dx = (1)/(4) + (1)/(4) = (2)/(4) = (1)/(2) The value of the integral is: (1)/(2) 15. Use the substitution t = ()/(2) to evaluate _0^()/(2) (4)/(3 + 5 ) d. Step 1: Apply the Weierstrass substitution formulas. Given t = ()/(2). Then d = (2)/(1+t^2) dt. And = (2t)/(1+t^2). Step 2: Change the limits of integration. When = 0, t = (0)/(2) = 0 = 0. When = ()/(2), t = (/2)/(2) = ()/(4) = 1. Step 3: Substitute these into the integral. _0^()/(2) (4)/(3 + 5 ) d = _0^1 (4)/(3 + 5 (2t)1+t^2) ((2)/(1+t^2)) dt = _0^1 (4)/(3(1+t^2) + 10t)1+t^2 · (2)/(1+t^2) dt = _0^1 (4(1+t^2))/(3+3t^2 + 10t) · (2)/(1+t^2) dt = _0^1 (8)/(3t^2 + 10t + 3) dt Step 4: Factor the denominator. 3t^2 + 10t + 3 = (3t+1)(t+3) Step 5: Use partial fraction decomposition. (8)/((3t+1)(t+3)) = (A)/(3t+1) + (B)/(t+3) 8 = A(t+3) + B(3t+1) Set t = -3: 8 = A(-3+3) + B(3(-3)+1) 8 = B(-8) B = -1. Set t = -(1)/(3): 8 = A(-(1)/(3)+3) + B(3(-(1)/(3))+1) 8 = A((8)/(3)) A = 3. So, the integrand becomes: (3)/(3t+1) - (1)/(t+3) Step 6: Evaluate the integral. _0^1 ( (3)/(3t+1) - (1)/(t+3) ) dt = [ (3 |3t+1|)/(3) - |t+3| ]_0^1 = [ |3t+1| - |t+3| ]_0^1 = [ |(3t+1)/(t+3)| ]_0^1 Now, apply the limits: = ((3(1)+1)/(1+3)) - ((3(0)+1)/(0+3)) = ((4)/(4)) - ((1)/(3)) = (1) - ((1)/(3)) = 0 - (- 3) = 3 The value of the integral is: 3 Drop the next question!