Quiz on Differentiation Rules

What is the Power Rule of Differentiation?

• d/dx(x^n) = n*x^(n-1)

• d/dx(x^n) = n*x^n

• d/dx(x^n) = n*x^(n+1)

• d/dx(x^n) = n/x^(n-1)

Answer explanation

The Power Rule states that the derivative of x raised to the power n is n times x raised to the power of (n-1). Thus, the correct choice is d/dx(x^n) = n*x^(n-1).

What does the Sum Rule of Differentiation state?

• f'(x) = u(x) - v(x)

• f'(x) = u(x) + v(x)

• f'(x) = u'(x) - v'(x)

• f'(x) = u'(x) + v'(x)

Answer explanation

The Sum Rule of Differentiation states that the derivative of a sum of functions is the sum of their derivatives. Therefore, f'(x) = u'(x) + v'(x) is the correct choice.

What is the Product Rule of Differentiation?

• f'(x) = u(x) / v(x)

• f'(x) = u'(x) * v(x) + u(x) * v'(x)

• f'(x) = u'(x) + v'(x)

• f'(x) = u(x) * v(x)

Answer explanation

The Product Rule states that the derivative of a product of two functions u(x) and v(x) is given by f'(x) = u'(x) * v(x) + u(x) * v'(x). This is the correct choice among the options.

What is the Quotient Rule of Differentiation?

• f'(x) = u * v'

• f'(x) = u / v

• f'(x) = u' + v'

• f'(x) = (u' * v - u * v') / v^2

Answer explanation

The Quotient Rule states that if f(x) = u/v, then the derivative f'(x) is given by (u' * v - u * v') / v^2. This formula accounts for the derivatives of both the numerator (u) and the denominator (v).

What is the Chain Rule of Differentiation?

• dy/dx = (dy/du) * (du/dx)

• dy/dx = dy/du + du/dx

• dy/dx = dy/du - du/dx

• dy/dx = du/dx / dy/du

Answer explanation

The Chain Rule states that the derivative of a composite function is the product of the derivatives of the outer and inner functions. Thus, dy/dx = (dy/du) * (du/dx) is the correct expression.

What is the derivative of x^5 using the Power Rule?

• 5x^4

• 5x^5

• x^4

• 4x^5

Answer explanation

Using the Power Rule, the derivative of x^n is n*x^(n-1). For x^5, n=5, so the derivative is 5*x^(5-1) = 5*x^4. Thus, the correct answer is 5x^4.

If f(x) = 6x^2 - 4x, what is f'(x)?

• 12x - 4

• 6x - 4

• 12x + 4

• 6x + 4

Answer explanation

To find f'(x), we differentiate f(x) = 6x^2 - 4x. The derivative of 6x^2 is 12x, and the derivative of -4x is -4. Thus, f'(x) = 12x - 4, which matches the correct answer choice.

Tags

CCSS.HSF.IF.A.2

What is the derivative of f(x) = x^2(x + 3) using the Product Rule?

• 3x^2 + 6x

• 2x^2 + 6x

• x^2 + 3

• 2x + 3

Answer explanation

Using the Product Rule, let u = x^2 and v = (x + 3). Then, f'(x) = u'v + uv' = (2x)(x + 3) + (x^2)(1) = 2x^2 + 6x + x^2 = 3x^2 + 6x. Thus, the correct answer is 3x^2 + 6x.

What is the derivative of f(x) = (x + 2)^3 / √x using the Quotient Rule?

• 2x + 3

• 3x^2 + 6x

• 5/2x^(3/2) + 9x^(1/2) - 4x^(-3/2)

• x^2 + 2

Answer explanation

Using the Quotient Rule, we differentiate f(x) = (x + 2)^3 / √x. The correct derivative simplifies to 5/2x^(3/2) + 9x^(1/2) - 4x^(-3/2), confirming that this is the right choice.

What is the derivative of f(x) = e^(sin(2x)) using the Chain Rule?

• 2cos(2x)e^(sin(2x))

• cos(2x)e^(sin(2x))

• e^(sin(2x))

• 2e^(sin(2x))

Answer explanation

Using the Chain Rule, we differentiate f(x) = e^(sin(2x)). The outer function is e^u, where u = sin(2x). The derivative is e^(sin(2x)) * cos(2x) * 2, leading to 2cos(2x)e^(sin(2x)), which is the correct choice.