Mastering Derivatives and Logarithms

What is the derivative of ln(x) with respect to x?

• 1

• 1/x

• ln(x)

• x

Apply the chain rule to find the derivative of ln(g(x)) where g(x) = x^2 + 3x.

• (2x + 3) / (x^2 + 3x)

• (2x + 3) / (x^2)

• (2x + 3) / (2x + 3)

• (x^2 + 3) / (2x + 3)

Differentiate ln(x^2 + 1) using the chain rule.

• 2/(x^2 + 1)

• x/(x^2 + 1)

• 2x/(x^2 + 1)

• 2x^2/(x^2 + 1)^2

What is the derivative of ln(2x + 1)?

• ln(2x + 1)

• 2/(2x + 1)

• 1/(2x + 1)

• 2x/(2x + 1)

If y = ln(x^2 + 4x + 4), what is dy/dx?

• (2x + 4) / (x^2 + 4x + 4)

• (x + 2) / (x^2 + 4)

• (2x + 2) / (x^2 + 4x)

• (4x + 4) / (x^2 + 4x + 4)

Differentiate ln(3x^3 + 2) using implicit differentiation.

• rac{3x^2}{3x^3 + 2}

• rac{27x^3}{3x^3 + 2}

• rac{9x^2}{6x^2}

• \frac{9x^2}{3x^3 + 2}

Find the derivative of ln(x + x^2 + 1) using the chain rule.

• (2x) / (x + x^2 + 1)

• (1 + 2x) / (x + x^2 + 1)

• (1 + x^2) / (x + x^2 + 1)

• (1 + x) / (x + x^2 + 1)

What is the second derivative of ln(x)?

• 1/x

• ln(x^2)

• -1/x^2

• -1/x

If y = ln(f(x)), what is dy/dx in terms of f(x) and f'(x)?

• dy/dx = f'(x) - f(x)

• dy/dx = f'(x) / f(x)

• dy/dx = f(x) + f'(x)

• dy/dx = f(x) * f'(x)

Differentiate ln(5x^2 + 3x) using the product rule.

• (10x^2 + 3) / (5x^2 + 3x^2)

• (5x + 3) / (5x^2 + 3x)

• (10x + 6) / (5x^2 + 3)

• (10x + 3) / (5x^2 + 3x)