Product and Quotient 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

What is the Quotient Rule in calculus?

Back

The Quotient Rule states that if you have two functions, f(x) and g(x), the derivative of their quotient is given by: \( \left( \frac{f}{g} \right)' = \frac{f'g - fg'}{g^2} \).

Front

Find the derivative of \( y = 2x^2(3x - 4) \).

Back

The derivative is \( 18x^2 - 16x \).

Front

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

Back

The derivative is \( -21x^6 + 6x^2 \).

Front

What is the formula for the derivative of a product of two functions?

Back

The formula is: \( (uv)' = u'v + uv' \).

Front

What is the formula for the derivative of a quotient of two functions?

Back

The formula is: \( \left( \frac{f}{g} \right)' = \frac{f'g - fg'}{g^2} \).

Front

Find the derivative of \( y(x) = \frac{4x^2 + 3}{2x^2 - 5} \).

Back

The derivative is \( y' = \frac{-52x}{(2x^2 - 5)^2} \).

Front

What is the significance of the derivative in calculus?

Back

The derivative represents the rate of change of a function with respect to its variable, indicating how the function behaves at any given point.

Front

How do you find the equation of the tangent line to a curve at a given point?

Back

To find the tangent line, calculate the derivative at the point to find the slope, then use the point-slope form of the line equation.

Front

Find the equation of the tangent line to the curve \( f(x) = (x - 2)(x + 1) \) at the point (1, 3).

Back

The equation is \( y = x + 2 \).