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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.
Write into the playground box as...
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)
The AI Calculus Solver for differential calculus is designed to handle:
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:
\[
f'(x) = 20x^3 – 9x^2 + 7.
\]
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:
\[
f'(x) = 2x \sin(x) + x^2 \cos(x).
\]
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:
\[
f'(x) = 3(3x^2 – 4x + 5)^2 (6x – 4).
\]
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:
\[
\frac{dy}{dx} = -\frac{x}{y}.
\]
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 \( x = 2 \).
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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Answer
Answer
Order of differentiation