f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1 and x = 3. Critical points are (1, f(1)) and (3, f(3)).

Using the power rule, f'(x) = -1/x^2.

Using the power rule, f'(x) = d/dx(3x^2) + d/dx(5x) - d/dx(7) = 6x + 5.

Using the power rule, f'(x) = 20x^3 - 3.

f'(x) = 2e^(2x). At x = 0, f'(0) = 2e^0 = 2.

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

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

Using the product rule, f'(x) = e^x * ln(x) + e^x/x. At x = 1, this simplifies to 0.

Using the product rule, f'(0) = e^0 * sin(0) + e^0 * cos(0) = 0 + 1 = 1.

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

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

f'(x) = cos(x) - sin(x), thus f'(π/4) = √2/2 - √2/2 = 0.

f'(x) = cos(x); f'(π/2) = cos(π/2) = 0.

f'(x) = sec^2(x); f'(0) = sec^2(0) = 1.

f'(x) = sec^2(x). At x = π/4, f'(π/4) = sec^2(π/4) = 2.

The derivative f'(x) = d/dx(tan(x)) = sec^2(x).

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

Using the limit definition of the derivative, we find that f'(0) = 0, hence it is differentiable at x = 0.

f'(x) = 3x^2 - 6x. At x = 2, f'(2) = 3(2^2) - 6(2) = 12 - 12 = 0.

Using the power rule, f'(x) = 3x^2 - 6x + 4.

Using the power rule, f'(x) = 3x^2 - 8x + 6.

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

This is a separable equation. Integrating gives ln|y| = 2x + C, hence y = Ce^(2x).

This is a separable equation. Integrating gives ln|y| = x + C, hence y = Ce^x.

The characteristic equation is r^2 - 5r + 6 = 0, giving roots 2 and 3. Thus, y = C1 e^(2x) + C2 e^(3x).

This is a first-order linear differential equation. The integrating factor is e^(-3x).

The characteristic equation is r^2 - 5r + 6 = 0, giving roots 2 and 3. Thus, y = C1 e^(2x) + C2 e^(3x).

The integral ∫(2x + 3)dx = x^2 + 3x + C.

The integral ∫(2x^3 - 4x + 1)dx = (1/2)x^4 - 2x^2 + x + C.

The integral of 1/x is ln|x| + C.