Basic Derivatives REVIEW

Front

What is the definition of a derivative?

Back

The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point.

Front

How do you find the equation of a tangent line to a function at a given point?

Back

To find the equation of the tangent line, calculate the derivative of the function to find the slope at the given point, then use the point-slope form of the line: y - y1 = m(x - x1).

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

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

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

How do you determine where the slope of the tangent line is positive?

Back

The slope of the tangent line is positive where the derivative of the function is greater than zero.

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

How do you find critical points of a function?

Back

Critical points occur where the derivative is zero or undefined. To find them, set the derivative equal to zero and solve for x.

Front

What is the significance of the first derivative test?

Back

The first derivative test helps determine whether a critical point is a local maximum, local minimum, or neither by analyzing the sign of the derivative before and after the critical point.

Front

What is the second derivative test?

Back

The second derivative test determines the concavity of a function at a critical point. If f''(x) > 0, the function is concave up (local minimum); if f''(x) < 0, it is concave down (local maximum).