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| Slope of the curve y=F(x) at the point P(c,f(c))is the number... |
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Definition
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How do you rationalize a function?
Under what situation do you attempt to rationalize? |
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Definition
When you have a two part numerator or denominatior, you put the numerator or denominator over itself, then change the conjugate (the sign that makes it a two part) and multiply it by the original function.
You attempt to rationalize when straight substitution gives you a meaningless result. |
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| What is the point slope form of a line? |
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Definition
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| What is the derivative of a function f at a point x? |
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Definition
f`(x)= [image]
The slope of the tangent line at a given point IS found by using the derivative.
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| What does the point slope formula deal with? |
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Definition
| Point slope form is used to work out or solve the equation of the tangent line. |
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| What does the derivative deal with? |
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Definition
It can give you the slope value of the tangent line. Only slope value. It won't give you the whole equation of the line. That is why you do derivative first. It equals M (see point slope formula).
This is used in rate of change problems.
Ex.
Find the slope of the tangent line.
Write the equation for the tangent line at that point. |
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| What is the derivative sum rule? |
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Definition
f(x)=5x5+3x2-2x-5
function: f(x)=5x5+3x2+(-2x)+(-5)
derivative f(x)=25x4+6x-2+0
sum of the derivatives
Once you have a function writen as a sum of terms, the derivative beecomes the sum of the DERIVATIVE OF THE TERMS.
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What is the chain Rule?
What does it allow us to accomplish? |
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Definition
If y=f(u) is a defferentiable function of u and u=g(x) is a differentiable function of x, then
y=f(g(x)) is a differentiable function of x and
dy/dx = dy/du*du/dx
OR
d/dx[f(g(x))] = f'(g(x)) g'(x)
Don't change inside
The Chain Rule allows us to differentiate composite functions.
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| What are the three pythagorean identities? |
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Definition
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| What are the chain rule formulas? |
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Definition
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| What is the derivative of a natural logarithm? |
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Definition
The derivative of the natural logarithm function is the reciprocal function.
When
f (x) = ln(x)
The derivative of f(x) is:
f ' (x) = 1 / x |
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Definition
logb(x y) = y ∙ logb(x)
For example:
log10(28) = 8 ∙ log10(2) |
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Definition
Constant Rule: The derivative of a constant is 0. That is f(x) = c is a horizontal line, with 0 slope. Alternatively, you can think of c as a function which never changes, so its rate of change is zero.
[image] |
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Definition
Scalar Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.
[image] |
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Definition
Sum Rule: The derivative of the sum (or difference) of two functions is the sum (or difference) of their derivatives.
[image] |
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Definition
Product Rule: The derivative of a product is "the first factor times the derivative of the second plus the second factor times the derivative of the first".
[image] |
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Definition
Quotient Rule: The derivative of a quotient is "the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, over the denominator squared".
[image]( |
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Definition
Chain Rule: The derivative of a function of u with respect to x is the derivative of the function with respect to u times the derivative of u with respect to x.
[image] |
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Definition
Absolute Value Rule: The derivative of the absolute value of u is the derivative of u multiplied by the ratio of u over the absolute value of u, wherever u is not equal to 0.
[image] , wherever u(x) ≠ 0. |
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| Logarithmic Quotient Rule |
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Definition
logb(x / y) = logb(x) - logb(y)
For example:
log10(3 / 7) = log10(3) - log10(7) |
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Derivative of natural logarithm |
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Definition
f (x) = ln(x)
The derivative of f(x) is:
f ' (x) = 1 / x |
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