Derivative Product Rule

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

Find the derivative of the function P(x) = (x^6 - 5)(x^4 + 5).

Back

P'(x) = 10x^9 + 30x^5 - 20x^3.

Front

How do you apply the Product Rule to find the derivative of a function?

Back

To apply the Product Rule, identify the two functions being multiplied, differentiate each function separately, and then combine them using the formula: (u*v)' = u'v + uv'.

Front

Let y = (x - 1)(x + 2). Find dy/dx.

Back

dy/dx = 2x + 1.

Front

What is the derivative of f(x) = (x - 1)(x + 2) evaluated at x = 0?

Back

f'(0) = 1.

Front

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

Back

y' = 18x^2 - 16x.

Front

Use the Product Rule to find the derivative of f(x) = -x^3(3x^4 - 2).

Back

f'(x) = -21x^6 + 6x^2.

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 is given by: (c*f(x))' = c*f'(x).

Front

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

Back

The derivative is f'(x) = n*x^(n-1).

Front

If u(x) = x^2 and v(x) = x^3, what is the derivative of their product?

Back

Using the Product Rule: (u*v)' = u'v + uv' = (2x)(x^3) + (x^2)(3x^2) = 2x^4 + 3x^4 = 5x^4.