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Computing Derivatives
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Calculus
Computing Derivatives
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Computing Derivatives Exercises
Constant Functions
The first kind of function is a constant function.If f(x) = C for some constant C, then f'(x) = 0.Constant functions have a slope of zero. On a graph, a constant function is a straight vertical lin...
Lines
We've seen lines before. The restroom at halftime, the blue things on notebook paper...and they were all over algebra. y = mx + b, anyone? Lines are a special case of polynomials....
Power Functions
A power function is any function of the form f(x) = xa, where a is any real number. Sample ProblemThe following are all power functions:Sample ProblemThe following are all power functions...
Exponential Functions
A power function has a variable x in the base and a constant for the power. An exponential function has a constant for the base and a variable for the power: f(x) = ax.In order to make life easier...
Logarithmic Functions
Logarithmic functions come in different flavors and bases. There's vanilla, strawberry...sugar cone, dish...but we'll start with the most basic one: f(x) = ln x.This function is defined for x > 0,...
Trigonometric Functions
The trigonometric functions are the functions used often in trigonometry:sin(x), cos(x), tan(x), csc(x), sec(x), and cot(x).We'll find the derivatives of some of these now, and some we'll leave unt...
Derivative of a Constant Multiple of a Function
Start with taking the derivatives of some lines.Sample ProblemFind the derivative of each function.f(x) = x g(x) = 3f(x) h(x) = 3g(x)Answer. These are all lines.The derivative...
Multiplication by 1
Remember that putting a negative sign in front of a function means the same thing as multiplying that function by 1.Sample QuestionLet g(x) = x2. Then we could think of this function as g(x) = (...
Fractions With a Constant Denominator
If we have a fraction where c is a constant, this is the same thing as . When asked to take derivatives of functions like this, the first thing we do is rewrite the original functio...
More Derivatives of Logarithms
Whenever possible, rewrite a function so it looks good before taking its derivative.Any logarithm logb a can be written asThis allows us to find derivatives of logarithms in bases besides e. W...
Derivative of a Sum (or Difference) of Functions
The derivative of a sum is the sum of the derivatives: (f + g)' = f' + g'. When the function has more than two terms, and some weird combination of addition and subtraction, the process is similar....
Derivative of a Product of Functions
Use the product rule whenever there is a function that's a product of two other functions of x. Iff(x) = ({something with an x}) × ({something else with an x}),then use the product rule to find f'...
Derivative of a Quotient of Functions
Division within derivatives is more complicated that the other rules we've seen so far. Make some space in the ol' memory bank for the Quotient Rule. The Quotient Rule states that the derivati...
Giving the Correct Answers
Some derivatives are simplifyable, while others aren't. We usually want to simplify the answer a little, but we don't want to do unnecessary work. How do we know when to stop simplifying?Soon, we'...
The Chain Rule
When dealing with a compositionh(x) = f(g(x)),the function f is called the outside function and the function g is called the inside function. f is the outside function because it's written on the o...
Derivatives of Inverse Trigonometric Functions
Finding derivatives of inverse trig functions will be similar to finding the derivative of ln x. Our main tool is the chain rule. We also need some background information:the fact that composing in...
The Chain Rule in Leibniz Notation
We stated the chain rule first in Lagrange notation. Since Leibniz notation lets us be a little more precise about what we're differentiating and what we're differentiating with respect to, we need...
Patterns
Combining the derivatives of basic functions with the chain rule gives us a lot of patterns that let us take derivatives of functions that seem complicated. Sample Problem Let h(x) = e{cos x}. If w...
Implicit Differentiation
Now it's time to throw a monkey wrench into the works, curve ball style. What happens when we mix two variables together, on both sides of the equation? Why calculus? Why?! We're entering the Twili...
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