Derivatives

Front

What is the derivative of a function?

Back

The derivative of a function measures how the function's output value changes as its input value changes. It represents the slope of the tangent line to the function's graph at a given point.

Front

Differentiate the function: f(x) = 2x + 3

Back

f'(x) = 2

Front

What is the power rule for differentiation?

Back

The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1), where n is a real number.

Front

Differentiate the function: f(x) = x^5

Back

f'(x) = 5x^4

Front

What is the product rule for differentiation?

Back

The product rule states that if u(x) and v(x) are functions, then the derivative of their product is given by (uv)' = u'v + uv'.

Front

Differentiate the function: f(x) = (2x + 2)(x - 5)^8

Back

f'(x) = 8(2x + 2)(x - 5)^7 + 2(x - 5)^8

Front

What is the quotient rule for differentiation?

Back

The quotient rule states that if u(x) and v(x) are functions, then the derivative of their quotient is given by (u/v)' = (u'v - uv')/v^2.

Front

Differentiate the function: f(x) = (2x^2 + 3)/(x)

Back

f'(x) = 2 + 3x^{-2}

Front

What does it mean for a derivative to be zero?

Back

A derivative being zero at a point indicates that the slope of the tangent line at that point is horizontal, which may correspond to a local maximum, minimum, or a point of inflection.

Front

Find the derivative of f(x) = -4x.

Back

f'(x) = -4