Mastering Derivatives and Functions

What is the derivative of f(x) = (3x^2 + 2x)^5 using the chain rule?

• f'(x) = 15(3x^2 + 2x)^4 * (6x)

• f'(x) = 5(3x^2 + 2x)^4 * (6x + 2)

• f'(x) = 5(3x^2 + 2x)^4 * (3x + 1)

• f'(x) = 5(3x^2 + 2x)^3 * (6x + 2)

If y = sin(2x^3), find dy/dx using the chain rule.

• dy/dx = 6x^2 * cos(2x^3)

• dy/dx = 12x^3 * cos(2x^3)

• dy/dx = 3x^2 * sin(2x^3)

• dy/dx = 2x^3 * cos(2x^3)

Determine if the function f(x) = x^3 - 3x is one-to-one.

• Yes

• It is one-to-one

• No

• It has a unique inverse

Find the derivative of the implicit function defined by x^2 + y^2 = 25.

• dy/dx = -y/x

• dy/dx = x/y

• dy/dx = 2x + 2y

• dy/dx = -x/y

Using implicit differentiation, find dy/dx for the equation x^2y + y^3 = 6.

• 2x^2y / (y^2 + 6)

• 3y^2 / (x^2 + 2xy)

• -2xy / (x^2 + 3y^2)

• -y^3 / (2x + 3y)

What is the derivative of g(t) = e^(3t^2) using the chain rule?

• g'(t) = 6t * e^(3t^2)

• g'(t) = 6 * e^(3t^2)

• g'(t) = 9t^2 * e^(3t)

• g'(t) = 3t * e^(3t^2)

Is the function f(x) = ln(x) + x^2 one-to-one? Justify your answer.

• No, the function f(x) = ln(x) + x^2 is not one-to-one.

• Yes, the function f(x) = ln(x) + x^2 is one-to-one only for x > 0.

• The function f(x) = ln(x) + x^2 is one-to-one because it is a polynomial.

• Yes, the function f(x) = ln(x) + x^2 is one-to-one.

Find the second derivative of h(x) = (2x + 1)^4 using the chain rule.

• h''(x) = 48(2x + 1)^2

• h''(x) = 32(2x + 1)^4

• h''(x) = 24(2x + 1)^2

• h''(x) = 16(2x + 1)^3

If y = x^2 + 2xy + y^2 = 10, find dy/dx using implicit differentiation.

• (x + y) / (x - y)

• (-2x - 2y) / (2x + 2y)

• (2x + 2y) / (-2x - 2y)

• (2x - 2y) / (2x + y)

Determine the derivative of the function f(x) = cos(5x^2) using the chain rule.

• 10x^2 sin(5x^2)

• 5x^2 sin(5x^2)

• -10x sin(5x^2)

• -5x cos(5x^2)