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: (u*v)' = u'v + uv'.

Front

What is the Quotient Rule in calculus?

Back

The Quotient Rule states that if you have two functions u(x) and v(x), the derivative of their quotient is given by: (u/v)' = (u'v - uv') / v^2.

Front

How do you apply the Product Rule to find the derivative of f(x) = x^2 * sin(x)?

Back

Using the Product Rule: f'(x) = (2x * sin(x)) + (x^2 * cos(x)).

Front

How do you apply the Quotient Rule to find the derivative of f(x) = (3x^2 + 1) / (x - 1)?

Back

Using the Quotient Rule: f'(x) = [(6x)(x - 1) - (3x^2 + 1)(1)] / (x - 1)^2.

Front

What is the Chain Rule in calculus?

Back

The Chain Rule is used to differentiate composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Front

How do you find the derivative of a composite function using the Chain Rule?

Back

To find the derivative of y = (3x + 2)^4, use the Chain Rule: dy/dx = 4(3x + 2)^3 * 3.

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 a function behaves at any given point.

Front

What is implicit differentiation?

Back

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y.

Front

How do you find the derivative of y^2 + x^2 = 1 using implicit differentiation?

Back

Differentiate both sides: 2y(dy/dx) + 2x = 0, then solve for dy/dx: dy/dx = -x/y.

Front

What is a critical point in calculus?

Back

A critical point occurs where the derivative of a function is zero or undefined, indicating potential local maxima or minima.