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 apply the Product Rule to find the derivative of f(x) = 2x^3 sin(x)?

Back

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

Front

What is the derivative of sin(x)?

Back

The derivative of sin(x) is cos(x).

Front

What is the derivative of cos(x)?

Back

The derivative of cos(x) is -sin(x).

Front

What is the derivative of x^n, where n is a constant?

Back

The derivative of x^n is nx^(n-1).

Front

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

Back

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

Front

What is the derivative of a constant multiplied by a function?

Back

If c is a constant and f(x) is a function, then the derivative of cf(x) is c * f'(x).

Front

How do you differentiate a polynomial function?

Back

To differentiate a polynomial function, apply the power rule to each term: if f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, then f'(x) = n*a_n x^(n-1) + (n-1)*a_(n-1) x^(n-2) + ... + a_1.

Front

What is the derivative of f(x) = -25x cos(x)?

Back

Using the Product Rule: Let u = -25x and v = cos(x). Then, u' = -25 and v' = -sin(x). So, f'(x) = u'v + uv' = -25cos(x) + 25xsin(x).

Front

What is the significance of the chain rule in differentiation?

Back

The chain rule is used to differentiate composite functions. If f(g(x)) is a composite function, then the derivative is f'(g(x)) * g'(x).