1. What is the python code for this problem:
# 1st: Import the sympy module import sympy as sp # Import sympy for symbolic mathematics # 2nd: Define the symbols m, n = sp.symbols('m n') # Define 'm' and 'n' as symbolic variables # 3rd: Define the function f(m) f = n**4 * (1 - 3*m)**6 # Define f(m) = n^4 * (1 - 3m)^6 # 4th: Compute the first derivative of f with respect to m f_prime = sp.diff(f, m) # First derivative f'(m) # 5th: Compute the second derivative of f with respect to m f_double_prime = sp.diff(f_prime, m) # Second derivative f''(m) # 6th: Display the results using pretty printing print("Function f(m):") sp.pprint(f) # Pretty print f(m) print("\nFirst derivative f'(m):") sp.pprint(f_prime) # Pretty print f'(m) print("\nSecond derivative f''(m):") sp.pprint(f_double_prime) # Pretty print f''(m)
2. What is the python code for this problem:
# 1st: Import the sympy module import sympy as sp # Import sympy for symbolic mathematics # 2nd: Define the symbol z = sp.symbols('z') # Define 'z' as a symbolic variable # 3rd: Define the function f(z) f = sp.sqrt(1 - 4*z) # Define the function f(z) = sqrt(1 - 4z) # 4th: Calculate the first derivative first_derivative = sp.diff(f, z) # Differentiate f(z) to find f'(z) # 5th: Calculate the second derivative second_derivative = sp.diff(first_derivative, z) # Differentiate f'(z) to find f''(z) # 6th: Calculate the third derivative third_derivative = sp.diff(second_derivative, z) # Differentiate f''(z) to find f'''(z) # 7th: Calculate the fourth derivative fourth_derivative = sp.diff(third_derivative, z) # Differentiate f'''(z) to find f''''(z) # 8th: Calculate the fifth derivative fifth_derivative = sp.diff(fourth_derivative, z) # Differentiate f''''(z) to find f'''''(z) # 9th: Display the results print("Function: f(z) =", f) # Output the function f(z) print("First derivative: f'(z) =", first_derivative) # Output the first derivative f'(z) print("Second derivative: f''(z) =", second_derivative) # Output the second derivative f''(z) print("Third derivative: f'''(z) =", third_derivative) # Output the third derivative f'''(z) print("Fourth derivative: f''''(z) =", fourth_derivative) # Output the fourth derivative f''''(z) print("Fifth derivative: f'''''(z) =", fifth_derivative) # Output the fifth derivative f'''''(z)
3. What is the python code for this problem:
# 1st: Import the sympy module import sympy as sp # Import sympy for symbolic mathematics # 2nd: Define the symbol z = sp.symbols('z') # Define 'z' as a symbolic variable # 3rd: Define the function m(z) m = 5*z**4 - 2*z**3 + z # Define the function m(z) = 5z^4 - 2z^3 + z # 4th: Calculate the first derivative first_derivative = sp.diff(m, z) # Differentiate m(z) to find m'(z) # 5th: Calculate the second derivative second_derivative = sp.diff(first_derivative, z) # Differentiate m'(z) to find m''(z) # 6th: Calculate the third derivative third_derivative = sp.diff(second_derivative, z) # Differentiate m''(z) to find m'''(z) # 7th: Calculate the fourth derivative fourth_derivative = sp.diff(third_derivative, z) # Differentiate m'''(z) to find m''''(z) # 8th: Display the results print("Function: m(z) =", m) # Output the function m(z) print("First derivative: m'(z) =", first_derivative) # Output the first derivative m'(z) print("Second derivative: m''(z) =", second_derivative) # Output the second derivative m''(z) print("Third derivative: m'''(z) =", third_derivative) # Output the third derivative m'''(z) print("Fourth derivative: m''''(z) =", fourth_derivative) # Output the fourth derivative m''''(z)
4. What is the python code for this problem:
# 1st: Import the sympy module import sympy as sp # Import sympy for symbolic mathematics # 2nd: Define the symbol n = sp.symbols('n') # Define 'n' as a symbolic variable # 3rd: Define the function f(n) f = n**2 / (1 - n) # Define the function f(n) = n^2 / (1 - n) # 4th: Calculate the first derivative first_derivative = sp.diff(f, n) # Differentiate f(n) to find f'(n) # 5th: Calculate the second derivative second_derivative = sp.diff(first_derivative, n) # Differentiate f'(n) to find f''(n) # 6th: Display the results print("Function: f(n) =", f) # Output the function f(n) print("First derivative: f'(n) =", first_derivative) # Output the first derivative f'(n) print("Second derivative: f''(n) =", second_derivative) # Output the second derivative f''(n)
5. What is the python code for this problem:
# 1st: Import the sympy module import sympy as sp # Import sympy for symbolic mathematics # 2nd: Define the symbols x, y = sp.symbols('x y') # Define 'x' and 'y' as symbolic variables # 3rd: Define the implicit function implicit_eq = 2*x**2 + 2*x*y + y**2 - 1 # Define the implicit equation 2x^2 + 2xy + y^2 - 1 # 4th: Differentiate implicitly to get dy/dx dy_dx = sp.diff(implicit_eq, y) / sp.diff(implicit_eq, x) # Differentiate implicitly to find dy/dx # 5th: Differentiate dy/dx to get the second derivative (d²y/dx²) d2y_dx2 = sp.diff(dy_dx, x) + sp.diff(dy_dx, y) * dy_dx # Differentiate dy/dx to find d²y/dx² # 6th: Differentiate d²y/dx² to get the third derivative (d³y/dx³) d3y_dx3 = sp.diff(d2y_dx2, x) + sp.diff(d2y_dx2, y) * dy_dx # Differentiate d²y/dx² to find d³y/dx³ # 7th: Simplify the third derivative d3y_dx3_simplified = sp.simplify(d3y_dx3) # Simplify the third derivative # 8th: Display the result print("Third derivative d³y/dx³:", d3y_dx3_simplified) # Output the third derivative d³y/dx³