Calculus

[calculusproject-blog]

Linear Approximation

The function f is differentiable at x = a. The equation of the line tangent to the graph of f at the point (a, f(a)) is:

[calculusproject-blog]

Related Rates of Change

When approaching a related rates question, follow these steps:

Read through the problem.

Draw a diagram

Write out three columns, entitled Given, Find, When.

Read through the problem again and fill out the columns appropriately.

Find the relating equation and find the derivative of both sides of the equation.

Plug in and solve.

[calculusproject-blog]

Implicit Differentiation

A function that is so complicated that an explicit rule for f(x) cannot be produced.

Use the chain rule to differentiate implicitly defined functions

[calculusproject-blog]

Natural Log

[calculusproject-blog]

The Chain Rule

[calculusproject-blog]

Trigonometric Functions

[calculusproject-blog]

Quotient Rule

[calculusproject-blog]

Product Rule

[calculusproject-blog]

Non-Normal Exponential Functions

The only function that is equal to its own derivative.

[calculusproject-blog]

Normal Exponential Functions

A normal exponential function:

Derivative:

OR

The derivative of an exponential function is proportional to the function itself.

OR

[calculusproject-blog]

f, f', and f''

The sign of the derivative of a function tells whether the original function is increasing or decreasing.

f' is negative = f is decreasing

f' is positive = f is increasing

f'' is negative = f' is decreasing, and f is concave down

f'' is positive = f' is increasing, and f is concave up

[calculusproject-blog]

Sum and Difference Rules

The derivative of a sum/difference of two functions is the sum/difference of their derivatives.

[calculusproject-blog]

Constant Multiple Rule

If a function is multiplied by a constant, the derivative is also.

[calculusproject-blog]

Power Rule

A normal power function:

Derivative:

[calculusproject-blog]

Constant Function Rule

A constant function is a linear function that has a slope of 0.

The derivative of a constant function is also 0.

A normal constant function:

Derivative:

[calculusproject-blog]

Linear Function Rule

The slope of a linear line is a constant.

The derivative of a linear line is also a constant.

A normal linear function:

Derivative:

[calculusproject-blog]

The value of the derivative at each a in the domain of f is defined as the limiting value of the average rate of change of f as h approaches 0, provided the limiting value exists.

This can be used to find the derivative formula of a given function.

Both symbols stand for the derivative