This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Another one ThegirlGodishelping🥰💙🫠 — let's solve it. Here are the derivatives for each function: 1. y = e^x^2x^3 + 1 Step 1: Identify u and v for the quotient rule (d)/(dx)((u)/(v)) = (u'v - uv')/(v^2). Let u = e^x^2 and v = x^3 + 1. Step 2: Find the derivatives of u and v. Using the chain rule for u: u' = e^x^2 · (d)/(dx)(x^2) = e^x^2 · (2x) = 2xe^x^2. For v: v' = 3x^2. Step 3: Apply the quotient rule. (dy)/(dx) = (2xe^x^2)(x^3 + 1) - (e^x^2)(3x^2)(x^3 + 1)^2 Step 4: Simplify the expression. (dy)/(dx) = e^x^2(2x(x^3 + 1) - 3x^2)(x^3 + 1)^2 (dy)/(dx) = e^x^2(2x^4 + 2x - 3x^2)(x^3 + 1)^2 (dy)/(dx) = e^x^2(2x^4 - 3x^2 + 2x)(x^3 + 1)^2 --- 3. y = sqrt(4x + 1)x^2 - 5 Step 1: Identify u and v for the quotient rule. Let u = sqrt(4x + 1) = (4x + 1)^1/2 and v = x^2 - 5. Step 2: Find the derivatives of u and v. Using the chain rule for u: u' = (1)/(2)(4x + 1)^-1/2 · (d)/(dx)(4x + 1) = (1)/(2)(4x + 1)^-1/2 · 4 = 2(4x + 1)^-1/2 = (2)/(sqrt(4x + 1)). For v: v' = 2x. Step 3: Apply the quotient rule. (dy)/(dx) = ((2)/(sqrt(4x + 1)))(x^2 - 5) - (sqrt(4x + 1))(2x)(x^2 - 5)^2 Step 4: Simplify the expression. Multiply the numerator by sqrt(4x+1) to clear the fraction in the numerator. (dy)/(dx) = (2(x^2 - 5))/(sqrt(4x + 1)) - 2xsqrt(4x + 1)(x^2 - 5)^2 · sqrt(4x + 1)sqrt(4x + 1) (dy)/(dx) = (2(x^2 - 5) - 2x(4x + 1))/((x^2 - 5)^2 sqrt(4x + 1)) (dy)/(dx) = (2x^2 - 10 - 8x^2 - 2x)/((x^2 - 5)^2 sqrt(4x + 1)) (dy)/(dx) = (-6x^2 - 2x - 10)/((x^2 - 5)^2 sqrt(4x + 1)) --- 5. f(x) = ((x^2))/(x) Step 1: Identify u and v for the quotient rule. Let u = (x^2) and v = x. Step 2: Find the derivatives of u and v. Using the chain rule for u: u' = -(x^2) · (d)/(dx)(x^2) = -(x^2) · (2x) = -2x(x^2). For v: v' = 1. Step 3: Apply the quotient rule. f'(x) = ((-2x(x^2))(x) - ((x^2))(1))/(x^2) Step 4: Simplify the expression. f'(x) = (-2x^2(x^2) - (x^2))/(x^2) f'(x) = -(2x^2(x^2) + (x^2))/(x^2) --- 7. y = ((x+1)^2)/((x-1)^2) Step 1: Identify u and v for the quotient rule. Let u = (x+1)^2 and v = (x-1)^2. Step 2: Find the derivatives of u and v. Using the chain rule for u: u' = 2(x+1) · (d)/(dx)(x+1) = 2(x+1) · 1 = 2(x+1). Using the chain rule for v: v' = 2(x-1) · (d)/(dx)(x-1) = 2(x-1) · 1 = 2(x-1). Step 3: Apply the quotient rule. (dy)/(dx) = ((2(x+1))(x-1)^2 - ((x+1)^2)(2(x-1)))/(((x-1)^2)^2) (dy)/(dx) = (2(x+1)(x-1)^2 - 2(x+1)^2(x-1))/((x-1)^4) Step 4: Factor out common terms from the numerator. (dy)/(dx) = (2(x+1)(x-1)[(x-1) - (x+1)])/((x-1)^4) (dy)/(dx) = (2(x+1)(x-1)[x-1-x-1])/((x-1)^4) (dy)/(dx) = (2(x+1)(x-1)(-2))/((x-1)^4) (dy)/(dx) = (-4(x+1)(x-1))/((x-1)^4) Step 5: Simplify by canceling (x-1). (dy)/(dx) = (-4(x+1))/((x-1)^3) --- 9. y = (1 + x)/(1 + x) Step 1: Identify u and v for the quotient rule. Let u = 1 + x and v = 1 + x. Step 2: Find the derivatives of u and v. For u: u' = x. For v: v' = - x. Step 3: Apply the quotient rule. (dy)/(dx) = (( x)(1 + x) - (1 + x)(- x))/((1 + x)^2) Step 4: Expand and simplify the numerator. (dy)/(dx) = ( x + ^2 x - (- x - ^2 x))/((1 + x)^2) (dy)/(dx) = ( x + ^2 x + x + ^2 x)/((1 + x)^2) Recall the identity ^2 x + ^2 x = 1. (dy)/(dx) = ( x + x + 1)/((1 + x)^2) (dy)/(dx) = (1 + x + x)/((1 + x)^2) Send me the next one 📸