Derivatives Review Fall 2024

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 graph of the function at a given point.

Front

Differentiate: f(x) = x^3 e^{4x}

Back

f '(x) = x^2 e^{4x}(4x + 3)

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).

Front

Differentiate: f(x) = 4x^3 - 10x^2 + 11x + 9

Back

f '(x) = 12x^2 - 20x + 11

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: y = tan(3x^2 + 2)

Back

y' = 6x sec^2(3x^2 + 2)

Front

What is the chain rule for differentiation?

Back

The chain rule states that if a function y = f(g(x)), then the derivative is given by dy/dx = f'(g(x)) * g'(x).

Front

Evaluate the limit: \lim_{h\rightarrow0}\left(\frac{5\left(x+h\right)^2-5x^2}{h}\right)

Back

10x

Front

What is the definition of a limit in calculus?

Back

A limit is a value that a function approaches as the input approaches some value. It is fundamental in defining derivatives and integrals.

Front

What is the tangent line approximation?

Back

The tangent line approximation uses the tangent line at a point to estimate the value of a function near that point. It is given by f(a) + f'(a)(x - a).