Derivatives - Product Rule

Front

What is the Product Rule in calculus?

Back

The Product Rule states that if you have two functions u(x) and v(x), the derivative of their product is given by: (uv)' = u'v + uv'.

Front

Find the derivative of f(x) = (x - 1)(x + 2).

Back

f'(x) = 2x + 1.

Front

Calculate f'(0) for f(x) = (x - 1)(x + 2).

Back

f'(0) = 1.

Front

Use the Product Rule to find the derivative of y = 2x^2(3x - 4).

Back

y' = 18x^2 - 16x.

Front

What is the derivative of a constant multiplied by a function?

Back

If c is a constant and f(x) is a function, then (cf)' = c * f'.

Front

If f(x) = x^2 and g(x) = x^3, find (fg)' using the Product Rule.

Back

(fg)' = f'g + fg' = 2x * x^3 + x^2 * 3x^2 = 5x^4.

Front

What is the derivative of y = x^2(x + 1)?

Back

y' = 2x(x + 1) + x^2(1) = 3x^2 + 2x.

Front

Find the derivative of y = (x^2 + 1)(x^3 - 2).

Back

y' = (2x)(x^3 - 2) + (x^2 + 1)(3x^2) = 2x^4 - 4x + 3x^2 + 3x^2 = 2x^4 + 6x^2 - 4x.

Front

What is the derivative of y = (3x^2)(4x^3)?

Back

y' = (6x)(4x^3) + (3x^2)(12x^2) = 24x^4 + 36x^4 = 60x^4.

Front

Explain how to apply the Product Rule step-by-step.

Back

1. Identify u and v. 2. Find u' and v'. 3. Apply the formula (uv)' = u'v + uv'.