Bibliothèque de solutions

The derivative of sin(2x² + 3x) is found using the chain rule, differentiating the outer sine function and the inner quadratic function separately, and then multiplying the results: cos(2x² + 3x) * (4x + 3).

The derivative of the function f(x) = 3x^2 - 4x + 7 is f'(x) = 6x - 4.

To find the area of the shaded region, subtract the area of a circle (formed by two semicircles) from the area of the enclosing rectangle.

The equation "\[\sqrt{(M-4)(M+4)} = 3\]" was solved by squaring both sides, expanding, simplifying, and taking the square root to find the solutions M = 5 and M = -5.

To find y when x = 4, substitute x = 4 into the equation y = 2x² - 3x + 1. Solving gives y = 21.

The limit of (1 + 1/x)^x as x approaches infinity is e.

The limit of sin(x)/x as x approaches 0 is 1, determined using L'Hôpital's rule.

The limit of (x² - 1) / (x - 1) as x approaches 1 is 2. This is found by factoring the numerator (difference of squares) and canceling the common factor of (x - 1).

The solution to the equation 25^x = 125 is x = 3/2. Squaring this value gives x^2 = 9/4.

The quadratic equation x^2 - x - 6 = 0 was solved by factoring. The solutions are x = 3 and x = -2.

The solution to 4 + 3 - 6 * (6 - 9) is 25.

The limit of (2x)/(x-7) as x approaches 7 from the right is positive infinity.

Solve the inequality 2x - 7 > 5. Find the solution to the inequality and learn how to isolate the variable. Answer: x > 6

Find the integral of x² + 3x - 2. Solution: (x³)/3 + (3x²)/2 - 2x + C

Calculate the definite integral of sin²(x) from 0 to π. Find the solution to this trigonometric integral problem, demonstrating the use of the identity sin²(x) = (1-cos(2x))/2. The answer is π/2.

Calculate the double integral ∫₀²∫₀¹ x²y dx dy. Step-by-step solution reveals the answer is 2/3.

Find the derivative of the function h(x) = ln(x² + 1) using the chain rule. The derivative is h'(x) = 2x / (x² + 1).

Learn how to find the derivative of the function g(x) = sin(x) * e^x using the product rule of differentiation. The solution reveals that the derivative is g'(x) = e^x (sin(x) + cos(x)).

Learn how to find the derivative of the function f(x) = x² + 3x - 2 using the power rule, constant multiple rule, and sum/difference rule. The derivative is f'(x) = 2x + 3.

Simplify the algebraic expression (a + b)² - (a - b)² using the difference of squares pattern. The solution reveals a simplified expression of 4ab.

Find the sum of the first 1000 natural numbers. This math problem can be solved using the formula for arithmetic series, resulting in a sum of 500,500.

Learn how to find the limit of the rational function (3x² + 2x) / (x² + 1) as x approaches infinity. The solution reveals a simple technique for evaluating limits at infinity and arrives at the answer: 3.

Find the derivative of the polynomial function f(x) = x³ + 2x² - 5x + 3 using the power rule. The derivative is f'(x) = 3x² + 4x - 5.

Learn how to solve the definite integral ∫₀² (x² + 2x) dx, step-by-step, and discover the answer: 20/3.

Find the derivative of the polynomial function f(x) = x³ + 2x² - 5x + 3 using the power rule. The derivative is f'(x) = 3x² + 4x - 5.

Solve the absolute value inequality |x - 3| ≤ 4. The solution is -1 ≤ x ≤ 7.

Solve for x in the inequality 2x - 7 > 5. The solution is x > 6.

Solve the system of equations 2x + y = 8 and x - y = 1. The solution is x = 3 and y = 2.

Solve the quadratic equation x² + 5x + 6 = 0. Factoring the equation, we find the solutions are x = -2 and x = -3.

Solve the equation 3x - 7 = 20. Find the value of x in this simple algebraic equation. The solution is x = 9.