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).

Tags

CCSS.HSF-IF.C.8B

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).