Colegio Marymount

Práctica Esencial # 1 de Derivadas — Clave para el Éxito en el Trimestral 2

Instrucción

Esta práctica es vital para tener éxito en el Trimestral 2. Resolver estos ejercicios con cuidado y mostrando cada paso fortalecerá tu comprensión de las derivadas. Recuerda: la práctica hace al maestro. Entre más practiques, mayor será tu confianza y precisión en el examen.

Find \(dy/dx\) or \(y'\) for the following:

  1. \(y = \dfrac{x^6 - 3x^2 + 5}{x^2}\)

  2. \(y = \dfrac{7}{\sqrt{(3x + 4)^4}}\)

  3. \(y = (5\sqrt{x} + 1)(2\sqrt{x} - 4)\)

  4. \(y = 8\sqrt{x}\left(\dfrac{1}{4\sqrt{x}} + \sqrt{x^2}\right)\)

  5. \(y = (3e^{2x + 5} - 1)^2\)

  6. \(y = e^{4x + 1} + 4e^{2 - x^2}\)

  7. \(y = 10 \ln(\sqrt{2x + 7})\)

  8. Find the gradient of the tangent to the curve \(y = e^{3x} - 2\) when \(x = 0\).

  9. Find the coordinates of the minimum point on the curve with equation \(y = e^{4x} - 8x\).

  10. Find the exact coordinates of the point on the curve \(y = \ln(x^2 + 8)\) where the gradient is \(1/3\).

Práctica Esencial # 2 de Derivadas

Exercise 4.1: Differentiating the Exponential Function

  1. Differentiate \(y = e^{4x}\).

  2. Calculate \(dy/dx\) for the function \(y = 5e^{2x} - 3\).

  3. Find the gradient of the tangent to the curve \(y = 2e^{x}\) at the point where \(x = 0\).

  4. Find the exact value of the gradient of the tangent to the curve \(y = e^{x} - 4x\) when \(x = 1\).

  5. Find the coordinates of the minimum point on the curve with equation \(y = e^{x} - 2x\).

  6. Differentiate the composite function \(y = (e^{x} + 1)^2\).

Exercise 4.2: Differentiating the Natural Logarithmic Function

  1. Differentiate the function \(y = \ln(3x)\).

  2. Find \(dy/dx\) for the function \(y = 4\ln(x + 2)\).

  3. Find the gradient of the tangent to the curve \(y = 2 + \ln x\) at the point \((1, 2)\).

  4. Find the gradient of the tangent to the curve \(y = \ln(x^2 + 4)\) when \(x = 1\).

  5. Find the exact coordinates of the point on the curve \(y = \ln(2x)\) where the gradient is \(1\).

  6. Find the derivative of the function \(y = \ln(2x - 1)\).

Exercise 4.3: Differentiating Products

  1. Use the product rule to find \(dy/dx\) when \(y = x^2 e^{x}\).

  2. Find the gradient of the tangent to the curve \(y = xe^{-x}\) at the point where \(x = 0\).

  3. Find the equation of the tangent to the curve \(y = x(x + 2)^2\) when \(x = 1\).

  4. Show that the derivative of \(y = (x + 2)^{x - 1}\) is \(2^{x - 1} \cdot 3x\).

  5. Find the exact coordinates of the turning point on the curve \(y = e^{x}(x - 2)\).

  6. Differentiate the product \(y = x \ln x\).

Exercise 4.4: Differentiating Quotients

  1. Use the quotient rule to find \(dy/dx\) for \(y = \dfrac{x + 2}{x}\).

  2. Find the values of \(x\) at the points where \(dy/dx = 0\) for the curve \(y = \dfrac{x - 1}{x^2}\).

  3. Find the equation of the normal to the curve \(y = \dfrac{x + 1}{x^2}\) at the point where \(x = 1\).

  4. Find the equation of the tangent to the curve \(y = \dfrac{x - 3}{2x}\) at the point where \(x = 4\).

  5. Show that the coordinates of the turning point on the curve \(y = \dfrac{x^2 + 1}{x - 1}\) are \((1 + \sqrt{2}, f(x))\).

  6. Differentiate the quotient \(y = \dfrac{x}{e^{x}}\).

Exercise 4.5: Differentiating Trigonometric Functions

  1. Find \(dy/dx\) for the function \(y = \sin(3x)\).

  2. Show that if \(y = \ln(\cos x)\), then \(dy/dx = -\tan x\).

  3. Calculate the derivative \(dy/dx\) for the function \(y = \cos(x^2 + 2x)\).

  4. Find the derivative \(dy/dx\) when \(y = 6\tan^2 x\).

  5. Differentiate the function \(y = \sin(5x)\) with respect to \(x\).

  6. Find the derivative of the product \(y = x^2 \cos x\).

Práctica Esencial # 3 de Derivadas — Hacia la Maestría

Instrucción

Esta práctica es vital para tener éxito en el Trimestral 2. Resolver estos ejercicios con cuidado y mostrando cada paso fortalecerá tu comprensión de las derivadas. Recuerda: la práctica hace al maestro. Entre más practiques, mayor será tu confianza y precisión en el examen.

Exercise 4.1: Differentiating Exponential Functions

  1. Find the derivative of the compound function \(y = e^{(x^3 - 4x)}\) by differentiating the exponent first.

  2. Calculate \(dy/dx\) for \(y = (e^{2x} + 5e^{-x})^4\) using the chain rule.

  3. Find the exact coordinates of the minimum point for the curve \(y = e^{2x} - 8e^{x} + 5\) by setting the derivative to zero.

Exercise 4.2: Differentiating Logarithmic Functions

  1. Differentiate \(y = \ln\left(\dfrac{x^3}{x^2 + 1}\right)\) after using the laws of logarithms to expand the fraction into a subtraction.

  2. Find \(dy/dx\) for \(y = \ln\left(\sqrt{5x^2 - 7x}\right)\) by moving the square root power \(\tfrac{1}{2}\) to the front of the logarithm first.

  3. Calculate the derivative of \(y = \ln(x^2 e^{4x})\) by simplifying the expression into individual terms before differentiating.

Exercise 4.3: Differentiating Products

  1. Use the product rule to differentiate \(y = e^{2x}(4x + 1)\), identifying your \(u\) and \(v\) terms clearly.

  2. Find the derivative of \(y = (x^2 + 3)\ln(2x - 1)\) and remember to apply the chain rule to the inner part of the logarithm.

  3. Find the equation of the tangent to the curve \(y = e^{x}\sin x\) at the point \((0, 0)\).

Exercise 4.4: Differentiating Quotients

  1. Use the quotient rule to find \(dy/dx\) for \(y = \dfrac{2x^2 + 1}{e^{4x}}\).

  2. Differentiate \(y = \dfrac{x^2}{\ln(x + 1)}\) and express your final answer as a single fraction.

  3. Calculate the derivative of \(y = \dfrac{x}{x^2 + 4}\) and simplify the resulting expression carefully.

Exercise 4.5: Differentiating Trigonometric Functions

  1. Find the derivative of \(y = \cos^4(3x)\) by applying the chain rule to the power, the cosine, and the angle.

  2. Calculate \(dy/dx\) for \(y = \tan(x^2 + 1)\) using the derivative of tangent \(\sec^2\) and the chain rule.

  3. Show that if \(y = \ln(\sin x)\), the derivative is \(dy/dx = \cot x\) by using the formula \(f'(x)/f(x)\).