Derivatives

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The Basic Differentiation Rules

Some differentiation rules are a snap to remember and use. These include the constant rule, power rule, constant multiple rule, sum rule, and difference rule. [more…]

How to Find the Derivative of a Line

The derivative is just a fancy calculus term for a simple idea that you probably know from algebra — slope. Slope is the fancy algebra term for steepness. And [more…]

How to Find the Derivative of a Curve

Calculus is the mathematics of change — so you need to know how to find the derivative of a parabola,which is a curve with a constantly changing slope. [more…]

How to Differentiate the Trigonometric Functions

You should memorize the derivatives of the six trig functions. Make sure you memorize the first two in the following list — they’re a snap. If you’re good at rote memorization, memorize the last four as [more…]

How to Differentiate Exponential and Logarithmic Functions

Differentiating exponential and logarithmic functions involves special rules. No worries — once you memorize a couple of rules, differentiating these functions is a piece of cake. [more…]

How to Find Derivatives Using the Product and Quotient Rules

There are special rules for finding the derivative of the product of two functions or the quotient of two functions; these are the product rule and the quotient rule, respectively. [more…]

How to Use Logarithmic Differentiation

For differentiating certain functions, logarithmic differentiation is a great shortcut. It spares you the headache of using the product rule or of multiplying the whole thing out and then differentiating [more…]

How to Differentiate Inverse Functions

There’s a difficult-looking formula involving the derivatives of inverse functions, but before you get to that, look at the following figure, which nicely sums up the whole idea. [more…]

How to Find High-Order Derivatives

Finding a second, third, fourth, or higher derivative is incredibly simple. The second derivative of a function is just the derivative of its first derivative. The third derivative is the derivative of [more…]

How to Know When a Derivative Doesn't Exist

There are three situations where a derivative fails to exist. The derivative of a function at a given point is the slope of the tangent line at that point. So, if you can’t draw a tangent line, there’s [more…]

How to Use the Chain Rule to Find the Derivative of Nested Functions

Sometimes, when you need to find the derivative of a nested function with the chain rule, figuring out which function is inside which can be a bit tricky — especially when a function is nested inside another [more…]

The Difference Quotient: The Bridge between Algebra (Slope) and Calculus (the Derivative)

One of the cornerstones of calculus is the difference quotient. The difference quotient — along with limits — allows you to take the regular old slope formula that you used to compute the slope of lines [more…]

How to Find a Function's Derivative by Using the Chain Rule

The chain rule is by far the trickiest derivative rule, but it’s not really that bad if you carefully focus on a few important points. [more…]

The Most Important Derivatives and Antiderivatives to Know

The table below shows you how to differentiate and integrate 18 of the most common functions. As you can see, integration reverses differentiation, returning the function to its original state, up to a [more…]

Calculus: How to Solve Differentiation Problems

In calculus, the way you solve a derivative problem depends on what form the problem takes. Common problem types include the chain rule; optimization; position, velocity, and acceleration; and related [more…]

How to Analyze Position, Velocity, and Acceleration with Differentiation

Every time you get in your car, you witness differentiation first hand. Your speed is the first derivative of your position. And when you step on the accelerator or the brake — accelerating or decelerating [more…]

How Coefficients Affect Differentiation

If the function you're differentiating begins with a coefficient, the coefficient has no effect on the process of differentiation. You just ignore it and differentiate according to the appropriate rule [more…]

How to Differentiate Implicitly

Sometimes you are asked to differentiate an equation that's not solved for y, like y5 + 3x2 = sin x – 4y3. This equation defines y implicitly as a function of [more…]

Compute the Slope of a Function Using the Difference Quotient

You can compute a function's slope by using the difference quotient. The difference quotient allows you to compute a slope if you don't initially have two points to plug into the slope formula. [more…]

How to Determine Maximum and Minimum Speeds of Moving Objects

One of the most practical uses of differentiation is finding the maximum or minimum values of a real-world function, for example, the maximum and minimum speeds of a moving object. [more…]

How to Find Changing Distance between Two Moving Objects

In a typical related rates problem, such as when you're finding a change in the distance between two moving objects, the rate or rates in the given information are constant, unchanging, and you have to [more…]

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