Differentiate y = e^x _e x using product rule. Find (dy)/(dx), if x2 + y2 + xy3 = 3 at the given point (1, 1). If y = (t2)/(1+t), x = (t)/(1+t), then find (dy)/(dx).

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Asked on September 8, 2026Mathematics

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.

Accepted Answer · 2026-09-08T16:44:08.641Z

Here are the solutions to the differentiation problems: (i). Differentiate y = (x^2 + 2)^5 Step 1: Apply the chain rule. Let u = x^2 + 2. Then y = u^5. (dy)/(dx) = (dy)/(du) · (du)/(dx) Step 2: Differentiate y with respect to u and u with respect to x. (dy)/(du) = 5u^4 (du)/(dx) = 2x Step 3: Substitute back u = x^2 + 2 and multiply the derivatives. (dy)/(dx) = 5(x^2 + 2)^4 · (2x) (dy)/(dx) = 10x(x^2 + 2)^4 (ii). Differentiate y = ( x)/( x) Step 1: Recognize that y = ( x)/( x) = x. Step 2: Differentiate y = x with respect to x. (dy)/(dx) = ^2 x (3a). Differentiate y = e^x _e x using product rule. Step 1: Identify u and v for the product rule, where y = uv. Let u = e^x and v = _e x = x. Step 2: Find the derivatives of u and v with respect to x. (du)/(dx) = e^x (dv)/(dx) = (1)/(x) Step 3: Apply the product rule formula: (dy)/(dx) = u (dv)/(dx) + v (du)/(dx). (dy)/(dx) = e^x ((1)/(x)) + ( x) e^x (dy)/(dx) = e^x ((1)/(x) + x) (3b). Find (dy)/(dx), if x^2 + y^2 + xy^3 = 3 at the given point (1, 1). Step 1: Differentiate both sides of the equation with respect to x using implicit differentiation. Remember to apply the product rule for xy^3. (d)/(dx)(x^2) + (d)/(dx)(y^2) + (d)/(dx)(xy^3) = (d)/(dx)(3) 2x + 2y (dy)/(dx) + (1 · y^3 + x · 3y^2 (dy)/(dx)) = 0 Step 2: Rearrange the equation to isolate (dy)/(dx). 2x + 2y (dy)/(dx) + y^3 + 3xy^2 (dy)/(dx) = 0 (2y + 3xy^2) (dy)/(dx) = -2x - y^3 (dy)/(dx) = (-2x - y^3)/(2y + 3xy^2) Step 3: Substitute the given point (x, y) = (1, 1) into the expression for (dy)/(dx). (dy)/(dx) = (-2(1) - (1)^3)/(2(1) + 3(1)(1)^2) (dy)/(dx) = (-2 - 1)/(2 + 3) = (-3)/(5) (dy)/(dx) = -(3)/(5) (3c). if y = (t^2)/(1+t), x = (t)/(1+t), then find (dy)/(dx) Step 1: Find (dy)/(dt) using the quotient rule. Let u = t^2 and v = 1+t. (dy)/(dt) = ((1+t)(2t) - (t^2)(1))/((1+t)^2) = (2t + 2t^2 - t^2)/((1+t)^2) = (t^2 + 2t)/((1+t)^2) Step 2: Find (dx)/(dt) using the quotient rule. Let u = t and v = 1+t. (dx)/(dt) = ((1+t)(1) - (t)(1))/((1+t)^2) = (1+t - t)/((1+t)^2) = (1)/((1+t)^2) Step 3: Apply the chain rule for parametric equations: (dy)/(dx) = (dy/dt)/(dx/dt). (dy)/(dx) = (t^2 + 2t)/((1+t)^2)(1)/((1+t)^2) Step 4: Simplify the expression. (dy)/(dx) = (t^2 + 2t) · ((1+t)^2)/((1+t)^2) = t^2 + 2t (dy)/(dx) = t^2 + 2t Send me the next one 📸

Accepted Answer · 2026-09-08T16:44:08.641Z

Here are the solutions to the differentiation problems: (i). Differentiate y = (x^2 + 2)^5 Step 1: Apply the chain rule. Let u = x^2 + 2. Then y = u^5. (dy)/(dx) = (dy)/(du) · (du)/(dx) Step 2: Differentiate y with respect to u and u with respect to x. (dy)/(du) = 5u^4 (du)/(dx) = 2x Step 3: Substitute back u = x^2 + 2 and multiply the derivatives. (dy)/(dx) = 5(x^2 + 2)^4 · (2x) (dy)/(dx) = 10x(x^2 + 2)^4 (ii). Differentiate y = ( x)/( x) Step 1: Recognize that y = ( x)/( x) = x. Step 2: Differentiate y = x with respect to x. (dy)/(dx) = ^2 x (3a). Differentiate y = e^x _e x using product rule. Step 1: Identify u and v for the product rule, where y = uv. Let u = e^x and v = _e x = x. Step 2: Find the derivatives of u and v with respect to x. (du)/(dx) = e^x (dv)/(dx) = (1)/(x) Step 3: Apply the product rule formula: (dy)/(dx) = u (dv)/(dx) + v (du)/(dx). (dy)/(dx) = e^x ((1)/(x)) + ( x) e^x (dy)/(dx) = e^x ((1)/(x) + x) (3b). Find (dy)/(dx), if x^2 + y^2 + xy^3 = 3 at the given point (1, 1). Step 1: Differentiate both sides of the equation with respect to x using implicit differentiation. Remember to apply the product rule for xy^3. (d)/(dx)(x^2) + (d)/(dx)(y^2) + (d)/(dx)(xy^3) = (d)/(dx)(3) 2x + 2y (dy)/(dx) + (1 · y^3 + x · 3y^2 (dy)/(dx)) = 0 Step 2: Rearrange the equation to isolate (dy)/(dx). 2x + 2y (dy)/(dx) + y^3 + 3xy^2 (dy)/(dx) = 0 (2y + 3xy^2) (dy)/(dx) = -2x - y^3 (dy)/(dx) = (-2x - y^3)/(2y + 3xy^2) Step 3: Substitute the given point (x, y) = (1, 1) into the expression for (dy)/(dx). (dy)/(dx) = (-2(1) - (1)^3)/(2(1) + 3(1)(1)^2) (dy)/(dx) = (-2 - 1)/(2 + 3) = (-3)/(5) (dy)/(dx) = -(3)/(5) (3c). if y = (t^2)/(1+t), x = (t)/(1+t), then find (dy)/(dx) Step 1: Find (dy)/(dt) using the quotient rule. Let u = t^2 and v = 1+t. (dy)/(dt) = ((1+t)(2t) - (t^2)(1))/((1+t)^2) = (2t + 2t^2 - t^2)/((1+t)^2) = (t^2 + 2t)/((1+t)^2) Step 2: Find (dx)/(dt) using the quotient rule. Let u = t and v = 1+t. (dx)/(dt) = ((1+t)(1) - (t)(1))/((1+t)^2) = (1+t - t)/((1+t)^2) = (1)/((1+t)^2) Step 3: Apply the chain rule for parametric equations: (dy)/(dx) = (dy/dt)/(dx/dt). (dy)/(dx) = (t^2 + 2t)/((1+t)^2)(1)/((1+t)^2) Step 4: Simplify the expression. (dy)/(dx) = (t^2 + 2t) · ((1+t)^2)/((1+t)^2) = t^2 + 2t (dy)/(dx) = t^2 + 2t Send me the next one 📸

Differentiate y = e^x _e x using product rule. Find (dy)/(dx), if x2 + y2 + xy3 = 3 at the given point (1, 1). If y = (t2)/(1+t), x = (t)/(1+t), then find (dy)/(dx).

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Differentiate y = e^x _e x using product rule. Find (dy)/(dx), if x2 + y2 + xy3 = 3 at the given point (1, 1). If y = (t2)/(1+t), x = (t)/(1+t), then find (dy)/(dx).

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