Derivatives: Product, Quotient & Chain Rule

Front

What is the Product Rule in differentiation?

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 differentiation?

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 differentiation?

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

Differentiate: f(x) = x^7 (5 + 8x)^3

Back

f '(x) = x^6 (5 + 8x)^2 (35 + 80x).

Front

Find the derivative of g(x)=(3x-2)/(x^2+2)

Back

g'(x) = (-3x^2 + 4x + 6)/(x^2 + 2)^2.

Front

Find y' for y= (2x+1)^{10}

Back

y' = 20(2x+1)^9.

Front

Differentiate: y = x^3 * cos(x)

Back

y' = 3x^2 * cos(x) - x^3 * sin(x).

Front

Find y' for y=(e^{5x}) + 1

Back

y' = 5e^{5x}.

Front

What is the derivative of a constant?

Back

The derivative of a constant is 0.

Front

What does it mean for a function to be differentiable?

Back

A function is differentiable at a point if it has a derivative at that point, meaning it is smooth and has no sharp corners or discontinuities.