Calculus Derivative Rules Practice

Answer explanation

To find the derivative of y with respect to x, we use the chain rule. The derivative of the outer function (u^2) is 2u, and the derivative of the inner function (x^3+1) is 3x^2. Applying the chain rule, we get dy/dx = 2u * 3x^2, where u = (x^3+1). Substituting u back in, we get dy/dx = 6x^2(x^3+1), which is the correct choice.

• 1

• -1
  

• 1/4

• -1/4

• 0

Answer explanation

To find the derivative of g(x) = (x-2)/(x+2), we can use the quotient rule: (v * du/dx - u * dv/dx) / v^2. Here, u = x-2 and v = x+2. After calculating the derivative, we plug in x = 2 and get g'(2) = 1/4, which is the correct answer.

Tags

CCSS.HSF.IF.A.2

Answer explanation

To find the derivative of the given function, we apply the power rule to each term. For the first term, the derivative is (1/2)x^(-1/2), and for the second term, it is (-4/3)x^(-7/3). Adding these derivatives together, we get the correct answer: (1/2)x^(-1/2) + (4/3)x^(-7/3).

• -2

• 1/6

• 1/2

• 2

• 6

Answer explanation

To find f'(2), we first need to find the derivative of f(x) = (x^2 - 2) / (x - 1). Using the quotient rule, we get f'(x) = (2x(x - 1) - (x^2 - 2)) / (x - 1)^2. Now, substitute x = 2 into the derivative: f'(2) = (4 - 2) / 1 = 2. So, the correct answer is 2.

Tags

CCSS.HSF.IF.A.2

• y = -7x - 3

• y = 7x + 7

• y = 7x + 11

• y = -5x - 1

• y = -5x - 5

Answer explanation

To find the tangent line, we need to find the derivative of the function f(x) = 4x^3 - 5x + 3. The derivative is f'(x) = 12x^2 - 5. At x = -1, f'(-1) = 7, which is the slope of the tangent line. The point on the graph is (-1, f(-1)) = (-1, 12). Using point-slope form, the equation of the tangent line is y = 7x + 11, which is the correct choice.

Answer explanation

To find the derivative of the given function y = (3 / (4 + x^2)), we can use the quotient rule: (dy/dx) = (v(du/dx) - u(dv/dx)) / v^2. Here, u = 3 and v = (4 + x^2). After applying the quotient rule and simplifying, we get the derivative as -6x / (4 + x^2)^2, which is the correct choice.

• 30x - 12

• 12x - 20

• 12x - 30

• 20x - 12

Answer explanation

To find the second derivative of the given function y = 5x^3 - 6x^2 + 4, first find the first derivative: dy/dx = 15x^2 - 12x. Then, find the second derivative by differentiating the first derivative: d^2y/dx^2 = 30x - 12. Thus, the correct answer is 30x - 12.

• -42

• 2

• 15

• 54

• 64

Answer explanation

The question asks to find the second derivative of the function y=3x^2+8x+4x^{-3} at x=1. By applying the power rule for differentiation twice and substituting x=1, we get the result as 54. Hence, the correct answer is 54.

Tags

CCSS.6.EE.A.2C