The derivative f'(x) = 3x^2 - 3. At x = 1, f'(1) = 3(1)^2 - 3 = 0, so the slope is 0.

The area between the curves is given by ∫(from -2 to 2) (4 - x^2) dx = [4x - x^3/3] from -2 to 2 = (8 - (8/3)) - (-8 + (8/3)) = 16 - (16/3) = 32/3.

The area is given by the integral from 1 to 2 of 1/x dx. This evaluates to [ln(x)] from 1 to 2 = ln(2) - ln(1) = ln(2).

The area is given by the integral from 0 to π/2 of cos(x) dx. This evaluates to [sin(x)] from 0 to π/2 = 1 - 0 = 1.

The area is given by the integral from 0 to π of sin(x) dx. This evaluates to [-cos(x)] from 0 to π = [1 - (-1)] = 2.

The area is given by the integral ∫_0^3 x^2 dx = [x^3/3]_0^3 = 27/3 = 9.

The area is ∫(1 to 3) x^2 dx = [1/3 * x^3] from 1 to 3 = (27/3 - 1/3) = 26/3.

The area under the curve y = x^3 from x = 1 to x = 2 is given by ∫(from 1 to 2) x^3 dx = [x^4/4] from 1 to 2 = (16/4) - (1/4) = 4 - 0.25 = 3.75.

The area under the curve y = x^4 from x = 0 to x = 1 is given by ∫(from 0 to 1) x^4 dx = [x^5/5] from 0 to 1 = 1/5.

Setting f'(x) = 3x^2 - 6x = 0 gives x(x - 2) = 0, so critical points are x = 0 and x = 2.

Setting f'(x) = 0 gives critical points at x = 1, 2, and 3.

f'(x) = 9x^2 - 5.

The derivative f'(x) = d/dx(5x^4 - 3x + 2) = 20x^3 - 3.

The derivative of e^x is e^x.

f'(x) = (2x)/(x^2 + 1), thus f'(0) = 0.

f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = (2*1)/(1^2 + 1) = 1.

Using the chain rule, f'(x) = (1/(x^2 + 1)) * (2x) = 2x/(x^2 + 1).

Using the derivatives of sine and cosine, f'(x) = cos(x) - sin(x).

Using the chain rule, f'(x) = cos(x^2) * 2x = 2x cos(x^2).

The derivative of tan(x) is sec^2(x).

Using the product rule, f'(x) = e^x * (x^2 + 2x).

f'(x) = 2x + 2.

f'(x) = 2x; f'(3) = 2*3 = 6.

The derivative f'(x) = d/dx(x^3 - 4x^2 + 6x) = 3x^2 - 8x + 6.

f'(x) = 4x^3.

The derivative f'(x) = d/dx(x^5 + 2x^3 - x) = 5x^4 + 6x^2 - 1.

The derivative f'(x) = d/dx(x^5) = 5x^4.

The left-hand derivative is -1 and the right-hand derivative is 1, hence the derivative at x = 0 is undefined.

f'(x) = 1/(2√x).

f'(x) = 2x + 2. At x = 1, f'(1) = 4. The tangent line is y - 3 = 4(x - 1) => y = 4x - 1.