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.