Product, Quotient and Chain Rules

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 a function that is the quotient of two functions u(x) and v(x), the derivative is given by: (u/v)' = (u'v - uv')/v^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 the derivative is given by: dy/dx = f'(g(x)) * g'(x).

Front

Find the derivative of f(x) = x^2 sin(x).

Back

f'(x) = 2x sin(x) + x^2 cos(x).

Front

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

Back

\frac{dy}{dx} = \frac{8x^4 - 60x^2}{(2x^2 - 5)^2}.

Front

What does the derivative represent in calculus?

Back

The derivative represents the slope of the tangent line to the curve at a given point.

Front

Find the derivative of f(x) = (-x^3 - 2)(-4x^2 - 1).

Back

f'(x) = 20x^4 + 3x^2 + 16x.

Front

What is the Power Rule in calculus?

Back

The Power Rule states that if f(x) = x^n, then f'(x) = n*x^(n-1).

Front

When do you use the Chain Rule?

Back

You use the Chain Rule when differentiating a function that is composed of another function, such as f(g(x)).

Front

What is the derivative of a constant?

Back

The derivative of a constant is zero.