Calculus - 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: (uv)' = u'v + uv'.

Front

How do you find the derivative of f(x) = xe^x using the Product Rule?

Back

To find the derivative of f(x) = xe^x, let u = x and v = e^x. Then, u' = 1 and v' = e^x. Applying the Product Rule: f'(x) = u'v + uv' = 1*e^x + x*e^x = e^x + xe^x.

Front

What is the derivative of a constant function?

Back

The derivative of a constant function is 0.

Front

How do you find the derivative of a polynomial function?

Back

To find the derivative of a polynomial function, apply the power rule: if f(x) = ax^n, then f'(x) = n*ax^(n-1).

Front

What is the power rule in calculus?

Back

The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1).

Front

How do you find the derivative of f(x) = (x - 1)(x + 2) at x = 0?

Back

First, find f'(x) using the Product Rule: f'(x) = (x - 1)'(x + 2) + (x - 1)(x + 2)'. Then evaluate f'(0) to get 1.

Front

What is the equation of the tangent line to a curve at a given point?

Back

The equation of the tangent line at a point (a, f(a)) is given by: y - f(a) = f'(a)(x - a).

Front

How do you find the slope of the tangent line to the curve f(x) = (x-2)(x+1) at the point (1,3)?

Back

First, find f'(x) using the Product Rule, then evaluate f'(1) to find the slope. The slope is 1, so the tangent line is y = x + 2.

Front

What is the derivative of f(x) = 8sin^2(x)?

Back

Using the chain rule and the power rule, f'(x) = 16sin(x)cos(x).

Front

What is the chain rule in calculus?

Back

The chain rule states that if you have a composite function f(g(x)), then the derivative is f'(g(x)) * g'(x).