Grade 12 Mathematics "Differential Calculus":

here are 10 questions based on Grade 12 Mathematics "Differential Calculus":

1. Differentiate the following function with respect to \( x \):
   \[
   f(x) = 3x^4 - 5x^2 + 2x - 7
   \]

2. Find the derivative of the function:
   \[
   g(x) = \sin(3x) + \cos(2x)
   \]

3. If \( h(x) = e^{2x} \cdot \ln(x) \), determine \( h'(x) \) using the product rule.

4. Given \( f(x) = \frac{2x^3 - 4x^2 + x}{x^2} \), simplify the function and then find \( f'(x) \).

5. Determine the critical points of the function:
   \[
   f(x) = x^3 - 6x^2 + 9x + 1
   \]

6. For the function \( f(x) = \ln(x^2 + 1) \), find the second derivative, \( f''(x) \).

7. Solve the differential equation:
   \[
   \frac{dy}{dx} = 5x^4 - 3x^2 + 2
   \]

8. Find the equation of the tangent line to the curve \( y = x^2 - 4x + 4 \) at the point \( (2, 0) \).

9. Determine the local maxima and minima for the function:
   \[
   f(x) = 2x^3 - 9x^2 + 12x - 4
   \]

10. Evaluate the following limit using L'Hôpital's rule:
    \[
    \lim_{{x \to 0}} \frac{\sin(x)}{x}
    \]

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