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About Differential Calculus AI

CalculusPop differential calculus AI solver that can handle a variety of differentiation problems. It can solve equations using first principles and the general rule for differentiation of functions like y = ax^n, sine, cosine, exponential functions, and logarithmic functions.

The calculus AI also covers methods of differentiation, including the differentiation of products and quotients, functions of functions, and implicit functions. It can handle applications of differentiation such as rates of change, velocity, acceleration, turning points, maximum and minimum values, points of inflexion, tangents and normals, and small changes in functions.

The AI can also differentiate parametric equations, common parametric equations, and hyperbolic functions. Moreso, it covers logarithmic differentiation and further logarithmic functions. With its ability to solve a wide range of differentiation problems, this AI is a valuable tool for students and professionals working in calculus and related fields.

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3x^2 \cdot \sin (2x) as 3x^2* sin (2x) 

1. \frac{d}{dx}(5x^2 + 3x - 7) as 5x^2 + 3x - 7

2. \frac{d}{dx}(e^{2x} + \sin(x)) as e^(2x) + sin(x)

3. \frac{d}{dx}(4x \cdot \cos(x)) as x * cos(x)

4. \frac{d}{dx}\left(\frac{3x^2}{x+1}\right) as 3x^2/ (x+1)

5. \frac{d}{dx}(\ln(2x)) as ln(2x)

6. \frac{d}{dx}(10x \cdot \tan(x)) as 10x * tan(x)

7. \frac{d}{dx}\left(\frac{x^2}{\sin(x)}\right) as x^2/sin(x)

8. \frac{d}{dx}(e^{3x}\cos(x)) as e^(3x)\cos(x)

9. \frac{d}{dx}(x^3 \cdot \ln(x)) as  x^3 * ln(x)

10. \frac{d}{dx}\left(\frac{2x+1}{x^2-3}\right) as (2x+1)/(x^2-3)

11. \frac{d}{dx}(5\sin(x) + 2\cos(x)) as 5*sin(x) + 2*cos(x)

12. \frac{d}{dx}(e^{4x} \cdot \tan(x)) as e^(4x) * tan(x)

13. \frac{d}{dx}\left(\frac{x^3}{\cos(x)}\right) as (x^3)/(cos(x))

14. \frac{d}{dx}(\ln(4x^2)) as ln(4x^2)

15. \frac{d}{dx}(7x^2 \cdot \sin(x)) as 7x^2 * sin(x)

16. \frac{d}{dx}\left(\frac{x}{e^x}\right) as  x/e^(x)

17. \frac{d}{dx}(3\cos(x) - 8\sin(x)) as 3*cos(x) - 8*sin(x)

18. \frac{d}{dx}(e^{x^2}\cdot \csc(x)) as e^(x^2)*csc(x)

19. \frac{d}{dx}(x^4 \cdot \log(x)) as  x^4 * log(x)

20. \frac{d}{dx}\left(\frac{3x-2}{x^3+1}\right) as (3x-2)/(x^3+1)

21.  x = \sin(t), y = \cos(t) as x = sin(t), y = cos(t)

22. y^2 - \ln(x) = 0 as y^2 - In(x)

23. \frac{d}{dx}\left( \ln(2x) \right) as ln(2x)

24. \frac{d}{dx}\left( \sinh(x) \right) as sinh(x)

25. \frac{d}{dx}\left( \cosh(x) \right) as cosh(x) 

26. \frac{d}{dx}\left( \tanh(x) \right) as tanh(x)

27. \frac{d}{dx}\left( \tanh(ax^2 + bx + c) \right) as tanh(ax^2 + bx + c)

28. \frac{d}{dx}\left( e^{\cos(x)} \right) as e^(cos(x))

29. \frac{d}{dx}\left( \log_a(x) \right) as log_a(x) 

30. \frac{d}{dx}\left( \sin^{-1}(x) \right) as sin^(-1)(x)

  1. Differentiation from first principles
  2. Differentiation of y = ax n by the general
    rule
  3. Differentiation of sine and cosine functions
  4. Differentiation of eax and ln ax
  5. Methods of differentiation
  6. Differentiation of common functions
  7. Differentiation of a product
  8. Differentiation of a quotient
  9. Function of a function
  10. Successive differentiation
  11. Some applications of differentiation
  12. Rates of change
  13. Velocity and acceleration
  14. Turning points
  15. Practical problems involving maximum
    and minimum values
  16. Points of inflexion
  17. Tangents and normals
  18. Small changes
  19. Differentiation of parametric equations
  20. Some common parametric equations
  21. Differentiation in parameters
  22. Differentiation of implicit functions
  23. Implicit functions
  24. Differentiating implicit functions
  25. Differentiating implicit functions containing products and quotients
  26. Logarithmic differentiation
  27. Differentiation of logarithmic functions
  28. Differentiation of further logarithmic
    functions
  29. Differentiation of [ f (x)]
  30. Differentiation of hyperbolic functions
  31. Standard differential coefficients of
    hyperbolic functions
  32. Differentiation of inverse trigonometric
    and hyperbolic functions
  33. Inverse functions
  34. Differentiation of inverse trigonometric
    functions
  35. Logarithmic forms of inverse hyperbolic
    functions
  36. Differentiation of inverse hyperbolic
    functions
  37. Partial differentiation
  38. Introduction to partial derivatives
  39. First-order partial derivatives
  40. Second order partial derivatives
  41. Total differential, rates of change and small
    changes
  42. Total differential
  43. Rates of change
  44. Small changes
  45. Maxima, minima and saddle points for
    functions of two variables

What Differential Calculus AI Can Solve

The AI Calculus Solver for differential calculus is designed to handle:

  1. Basic Derivatives: Differentiation of polynomial, trigonometric, exponential, and logarithmic functions.
  2. Advanced Differentiation Techniques:
    • Product rule, quotient rule, and chain rule.
    • Implicit differentiation.
    • Higher-order derivatives.
  3. Applications of Derivatives:
    • Tangents and normals to curves.
    • Optimization problems (finding maxima and minima).
    • Motion analysis (velocity and acceleration).
  4. Real-World Problems:
    • Related rates.
    • Growth and decay problems.

 

1. Basic Differentiation

Problem:
Find the derivative of \( f(x) = 5x^4 – 3x^3 + 7x – 2 \).

Solution:
1. Identify the Rule: Use the power rule, which states \( \frac{d}{dx}[x^n] = n x^{n-1} \).
2. Differentiate Each Term:
– \( \frac{d}{dx}[5x^4] = 20x^3 \),
– \( \frac{d}{dx}[-3x^3] = -9x^2 \),
– \( \frac{d}{dx}[7x] = 7 \),
– \( \frac{d}{dx}[-2] = 0 \).
3. Combine Results:
\[
f'(x) = 20x^3 – 9x^2 + 7.
\]

Answer:
\[
f'(x) = 20x^3 – 9x^2 + 7.
\]

 

2. Product Rule

Problem:
Find the derivative of \( f(x) = x^2 \sin(x) \).

Solution:
1. Identify the Rule: Use the product rule, \( \frac{d}{dx}[u v] = u’ v + u v’ \), where:
– \( u = x^2 \) and \( v = \sin(x) \).
2. Differentiate \( u \) and \( v \):
– \( u’ = \frac{d}{dx}[x^2] = 2x \),
– \( v’ = \frac{d}{dx}[\sin(x)] = \cos(x) \).
3. Apply the Product Rule:
\[
f'(x) = u’ v + u v’ = 2x \sin(x) + x^2 \cos(x).
\]

Answer:
\[
f'(x) = 2x \sin(x) + x^2 \cos(x).
\]

 

3. Chain Rule

Problem:
Find the derivative of \( f(x) = (3x^2 – 4x + 5)^3 \).

Solution:
1. Identify the Rule: Use the chain rule, \( \frac{d}{dx}[f(g(x))] = f'(g(x)) g'(x) \), where:
– Outer function: \( f(u) = u^3 \),
– Inner function: \( g(x) = 3x^2 – 4x + 5 \).
2. Differentiate the Outer Function:
– \( \frac{d}{du}[u^3] = 3u^2 \).
3. Differentiate the Inner Function:
– \( \frac{d}{dx}[3x^2 – 4x + 5] = 6x – 4 \).
4. Apply the Chain Rule:
\[
f'(x) = 3(3x^2 – 4x + 5)^2 \cdot (6x – 4).
\]

Answer:
\[
f'(x) = 3(3x^2 – 4x + 5)^2 (6x – 4).
\]

 

4. Implicit Differentiation

Problem:
Find \( \frac{dy}{dx} \) if \( x^2 + y^2 = 25 \).

Solution:
1. Differentiate Both Sides with Respect to \( x \):
– \( \frac{d}{dx}[x^2] = 2x \),
– \( \frac{d}{dx}[y^2] = 2y \frac{dy}{dx} \),
– \( \frac{d}{dx}[25] = 0 \).
So,
\[
2x + 2y \frac{dy}{dx} = 0.
\]
2. Solve for \( \frac{dy}{dx} \):
\[
2y \frac{dy}{dx} = -2x,
\]
\[
\frac{dy}{dx} = -\frac{x}{y}.
\]

Answer:
\[
\frac{dy}{dx} = -\frac{x}{y}.
\]

 

5. Optimization Problem

Problem:
Find the maximum value of \( f(x) = -x^2 + 4x + 5 \).

Solution:
1. Find the Critical Points:
– Differentiate \( f(x) \):
\[
f'(x) = -2x + 4.
\]
– Set \( f'(x) = 0 \) to find critical points:
\[
-2x + 4 = 0 \quad \Rightarrow \quad x = 2.
\]
2. Determine if it’s a Maximum or Minimum:
– Use the second derivative:
\[
f”(x) = -2.
\]
– Since \( f”(x) < 0 \), \( f(x) \) has a maximum at \( x = 2 \).
3. Find the Maximum Value:
– Substitute \( x = 2 \) into \( f(x) \):
\[
f(2) = -(2)^2 + 4(2) + 5 = -4 + 8 + 5 = 9.
\]

Answer:
The maximum value is \( 9 \) at \( x = 2 \).

 

6. Related Rates

Problem:
A spherical balloon is being inflated so that its volume increases at a rate of \( 100 \, \text{cm}^3/\text{s} \). How fast is the radius increasing when the radius is \( 10 \, \text{cm} \)?

Solution:
1. Write the Formula for the Volume of a Sphere:
\[
V = \frac{4}{3} \pi r^3.
\]
2. Differentiate with Respect to Time \( t \):
\[
\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}.
\]
3. Substitute Known Values:
– \( \frac{dV}{dt} = 100 \),
– \( r = 10 \).
So,
\[
100 = 4 \pi (10)^2 \frac{dr}{dt}.
\]
4. Solve for \( \frac{dr}{dt} \):
\[
100 = 400 \pi \frac{dr}{dt} \quad \Rightarrow \quad \frac{dr}{dt} = \frac{100}{400\pi} = \frac{1}{4\pi}.
\]

Answer:
The radius is increasing at a rate of \( \frac{1}{4\pi} \, \text{cm/s} \).

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Order of differentiation