Calculus Product Rule Derivatives

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

How do you differentiate the function f(x) = -25xcos(x) using the Product Rule?

Back

To differentiate f(x) = -25xcos(x), let u = -25x and v = cos(x). Then, u' = -25 and v' = -sin(x). Applying the Product Rule: f'(x) = u'v + uv' = (-25)(cos(x)) + (-25x)(-sin(x)) = 25xsin(x) - 25cos(x).

Front

What is the derivative of f(x) = 5cos^2(x)?

Back

Using the chain rule and the Product Rule, the derivative is f'(x) = -10sin(x)cos(x).

Front

How do you apply the Product Rule to the function f(x) = 2x^3sin(x)?

Back

Let u = 2x^3 and v = sin(x). Then, u' = 6x^2 and v' = cos(x). Applying the Product Rule: f'(x) = u'v + uv' = (6x^2)(sin(x)) + (2x^3)(cos(x)).

Front

What is the general formula for the derivative of a power function f(x) = x^n?

Back

The derivative of a power function is given by f'(x) = nx^(n-1).

Front

What is the chain rule in calculus?

Back

The chain rule is a formula for computing the derivative of the composition of two or more functions. If f(g(x)) is a composite function, then the derivative is f'(g(x)) * g'(x).

Front

How do you differentiate a function that is a product of three functions, such as f(x) = u(x)v(x)w(x)?

Back

To differentiate a product of three functions, use the extended Product Rule: f'(x) = u'vw + uv'w + uvw'.

Front

What is the derivative of f(x) = x^3 + 2x^2 - 5x + 7?

Back

The derivative is f'(x) = 3x^2 + 4x - 5.

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 as its input changes.

Front

How do you find the critical points of a function using its derivative?

Back

Critical points occur where the derivative is zero or undefined. Set f'(x) = 0 and solve for x to find critical points.