1. 2 Understanding The Derivative Ap Calculus Multiple Choice
2. 2 Understanding The Derivativeap Calculus Solver
3. 2 Understanding The Derivative Ap Calculus 14th Edition
4. 2 Understanding The Derivative Ap Calculus Integrals

You know you’ll have to know your derivatives inside and out in order to score high on the AP Calculus exam. But what are the most common derivatives you’ll see on the test? In this short article, we’ll let you in on the secret!

Know the Basics

Asked. 09/29/19 Calculus Derivatives. Find the y-intercept of the tangent line at the point P (-2, f(-2)) on the graph of. F(x) = x^2 + 2x + 3. If we recall the definition of a derivative of a function at a point, one of the definitions is. If we compare this definition to the limit we see that that this is the limit definition of a derivative, so we need to find the function and the point at which we are evaluating the derivative at. 2.1 Average Rate of Change. 2.2 Definition of the Derivative. 2.3 Differentiability.

Math AP®︎/College Calculus AB Differentiation: composite, implicit, and inverse functions Calculating higher-order derivatives. Calculating higher-order derivatives. The big idea of differential calculus is the concept of the derivative, which essentially gives us the direction, or rate of change, of a function at any of its points. Learn all about derivatives and how to find them here.

First of all, it’s very important to have the basics down. That includes: the Power Rule, Product Rule, Quotient Rule, and Chain Rule, among others.

Don’t forget about those special functions either. You’ll have to memorize the derivative rules for trig, exponential, and logarithmic functions.

Derivatives of trigonometric, exponential, and logarithmic functions

Less common, but no less important are the rules for inverse trig functions.

Check out Calculus Review: Derivative Rules and Derivatives on the AP Calculus AB & BC Exams: A Refresher for more.

Common Derivatives

Now let’s take a look at a few problems involving common derivatives that are modeled after actual AP Calculus problems.

Problem 1

Find The tangent line to the curve f(x) = x⁴ + 3x – 10 at the point (1, -6).

Solution

To find a tangent line, first take the derivative. Here, we have to use the Power Rule and Sum/Difference Rule.

f '(x) = 4x³ + 3

Then you can find the slope and the equation of the tangent line.

Slope at x = 1: f '(1) = 4(1)³ + 3 = 7

y – (-6) = 7(x – 1)

y = 7x – 13.

Problem 2

The position of a particle moving along the x-axis at time t is x(t) = sin( cos( 4t ) ), for 0 ≤ t ≤ π. Find
the velocity at time t = π/8.

Solution

Find velocity by taking the derivative of the position function. Be careful — we need two applications of the Chain Rule for this one!

v(t) = x '(t) = cos( cos( 4t ) ) ( -sin( 4t ) ) (4)

At time t = π/8, the velocity is equal to:

v(π/8) = cos( cos( 4(π/8) ) ) ( -sin( 4(π/8) ) ) (4) = cos( cos( π/2 ) ) ( -sin( π/2 ) ) (4)

= cos(0)(-1)(4) = (1)(-1)(4) = -4.

Problem 3

Find all inflection points of the curve defined by .

Solution

You can find inflection points by taking the second derivative. Of course, you must take the first derivative first. Don’t forget to rewrite your radical as a power and use Chain Rule.

Use Quotient Rule to help find the second derivative.

After simplification, we find that the second derivative is never equal to 0 and never undefined. Therefore this function has no inflection points.

Problem 4

Solution

You’ll need both the Product and Chain Rules for this one.

Problem 5

Suppose g(x) = e^4x + 5. At what value of x is the slope of the tangent line to g(x) equal to 3?

Solution

Again, the slope of the tangent line is equal to a derivative value.

g '(x) = 4e^4x

Set g '(x) = 3 and solve:

4e^4x = 3

e^4x = 3/4

4x = ln(3/4)

x = ln(3/4) / 4 ≈ -0.0719

Summary

The five problems above represent just a small sampling of what you’ll find on an AP Calculus AB or BC exam. Nevertheless, the majority of problems involving derivatives do tend to fall into these basic formats.

• Know your basic rules, especially the Chain Rule.
• Memorize the derivatives of the special functions.
• And above all, know which tools apply in each situation.

Good luck on the exam!

About Shaun Ault

Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!

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It is all about slope!

Let us Find a Derivative!

To find the derivative of a function y = f(x) we use the slope formula:

Slope = Change in YChange in X = ΔyΔx

And (from the diagram) we see that:

Now follow these steps:

• Fill in this slope formula: ΔyΔx = f(x+Δx) − f(x)Δx
• Simplify it as best we can
• Then make Δx shrink towards zero.

Like this:

Example: the function f(x) = x²

We know f(x) = x², and we can calculate f(x+Δx) :

Result: the derivative of x² is 2x

In other words, the slope at x is 2x

We write dx instead of 'Δx heads towards 0'.

And 'the derivative of' is commonly written :

x² = 2x
'The derivative of x² equals 2x'
or simply 'd dx of x² equals 2x'

What does x² = 2x mean?

It means that, for the function x², the slope or 'rate of change' at any point is 2x.

So when x=2 the slope is 2x = 4, as shown here:

Or when x=5 the slope is 2x = 10, and so on.

Note: sometimes f’(x) is also used for 'the derivative of':

f’(x) = 2x
'The derivative of f(x) equals 2x'
or simply 'f-dash of x equals 2x'

Let's try another example.

Example: What is x³ ?

We know f(x) = x³, and can calculate f(x+Δx) :

Start with:

f(x+Δx) = (x+Δx)3

Expand (x + Δx)3:

2 Understanding The Derivative Ap Calculus Multiple Choice Have a play with it using the Derivative Plotter. Derivatives of Other Functions We can use the same method to work out derivatives of other functions (like sine, cosine, logarithms, etc). But in practice the usual way to find derivatives is to use: Example: what is the derivative of sin(x) ? On Derivative Rules it is listed as being cos(x) Done. Using the rules can be tricky! Example: what is the derivative of cos(x)sin(x) ? You can't just find the derivative of cos(x) and multiply it by the derivative of sin(x) ... you must use the 'Product Rule' as explained on the Derivative Rules page. It actually works out to be cos2(x) − sin2(x) So that is your next step: learn how to use the rules. 2 Understanding The Derivativeap Calculus Solver Notation 'Shrink towards zero' is actually written as a limit like this: 'The derivative of f equals the limit as Δx goes to zero of f(x+Δx) - f(x) over Δx' Or sometimes the derivative is written like this (explained on Derivatives as dy/dx): 2 Understanding The Derivative Ap Calculus 14th Edition The process of finding a derivative is called 'differentiation'. 2 Understanding The Derivative Ap Calculus Integrals Where to Next? Go and learn how to find derivatives using Derivative Rules, and get plenty of practice: