derivative review

Front

What is the definition of a derivative?

Back

The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero. It represents the slope of the tangent line to the function at that point.

Front

How do you find the derivative of a constant function?

Back

The derivative of a constant function is always zero.

Front

What is implicit differentiation?

Back

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y, rather than explicitly solving for y.

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 constant.

Front

How do you find the equation of the normal line to a curve at a given point?

Back

To find the equation of the normal line, first find the slope of the tangent line at the point, then take the negative reciprocal of that slope to get the slope of the normal line. Use the point-slope form of a line to write the equation.

Front

What is the derivative of e^x?

Back

The derivative of e^x is e^x.

Front

What is the product rule for differentiation?

Back

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

Front

What is the chain rule for differentiation?

Back

The chain rule states that if a function y is composed of two functions u and x, then the derivative of y with respect to x is dy/dx = dy/du * du/dx.

Front

How do you differentiate a function involving trigonometric functions?

Back

To differentiate trigonometric functions, use the specific derivatives: d/dx(sin(x)) = cos(x), d/dx(cos(x)) = -sin(x), d/dx(tan(x)) = sec^2(x), etc.