Mastering Derivatives and Applications

What is the derivative of f(x) = 3x^2 + 5x - 7?

• 9x + 5

• 5x + 3

• 6x - 7

• 6x + 5

Find the derivative of g(t) = sin(t) + cos(t).

• g'(t) = sin(t) + cos(t)

• g'(t) = cos(t) - sin(t)

• g'(t) = -sin(t) - cos(t)

• g'(t) = -cos(t) + sin(t)

If h(x) = e^x, what is h'(x)?

• x * e^x

• e^x + 1

• e^x

• e^(x+1)

Determine the derivative of f(x) = ln(x^2 + 1).

• f'(x) = 1/(x^2 + 1)

• f'(x) = 2x/(x^2 + 1)

• f'(x) = 2/(x^2 + 1)

• f'(x) = 2x^2/(x^2 + 1)

What is the second derivative of f(x) = x^3 - 3x + 2?

• 6x

• 3x^2 - 3

• x^2 - 3

• 6x^2 - 3

Find the critical points of f(x) = x^4 - 4x^2.

• x = 1

• x = -1

• x = 2

• x = 0, x = sqrt(2), x = -sqrt(2)

Calculate the derivative of f(x) = 5/x^2.

• 10/x^2

• -5/x^3

• 5/x^4

• -10/x^3

If f(x) = x^2 + 2x, find the equation of the tangent line at x = 1.

• y = 4x - 1

• y = 2x + 3

• y = x^2 + 2

• y = 5x - 2

What is the derivative of f(x) = tan(x)?

• sin(x)

• cos(x)

• sec(x)

• sec^2(x)

Determine the maximum and minimum values of f(x) = -x^2 + 4x on the interval [0, 4].

• Maximum: 10 at x = 3; Minimum: -1 at x = 1

• Maximum: 6 at x = 0; Minimum: 1 at x = 3

• Maximum: 8 at x = 2; Minimum: 0 at x = 0 and x = 4

• Maximum: 4 at x = 1; Minimum: 2 at x = 2