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📖 = included in formula booklet • 🚫 = not in formula booklet
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Differentiation
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
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📖 = included in formula booklet • 🚫 = not in formula booklet
The limit x→alimf(x) is the value f(x) approaches as x approaches a.
The IB may test your understanding of the gradient of the curve as the limit of
as (x2−x1) goes to zero.
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Given a table of values:
The derivative of f(x) is denoted f′(x) and is given by
For a curve y=f(x), f′(x) is the gradient or slope.
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You can graph f′(x) using the following steps:
Press the Y= key.
In one of the available function lines (e.g. Y_1), enter the expression for f(x).
In another available line (e.g. Y_2), input the derivative function usingMATH then 8:nDeriv( in the following format:
To enter Y1, press VARS then scroll to Y-VARS and select FUNCTION then Y1.
Press GRAPH to display both the original graph f and the derivative f′.
The graph of f′ may take a little bit longer depending on the original function.
After graphing f′, you may use all the other graphing functions on the calculator (intersect, zero, and value).
dxdy is the rate of change of y with respect to x. That is, dxdy tells us how much y changes in response to a change in x.
If y=f(x), then dxdy=f′(x).
The product and quotient rules are given by
L:mx+c is tangent to f(x) at x=a means
Using point slope form the equation of the tangent is:
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The normal to f(x) at x=a is the line that passes through (a,f(a)) and is perpendicular to the tangent:
Using point slope form the equation of the tangent is:
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Stationary points are often local extrema.
If f′(a)=0, f is decreasing to the left of a (f′(x)<0), and f is increasing to the right of a (f′(x)>0), then (a,f(a)) is a local minimum.
If f′(a)=0, f is increasing to the left of a (f′(x)<0), and f is decreasing to the right of a (f′(x)>0), then (a,f(a)) is a local maximum.
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Optimisation problems require you to find a minimum or maximum value by producing a function f(x), taking its derivative, solving f′(x)=0, and confirming which stationary point(s) are minima or maxima.
The derivative of the derivative of a function is its second derivative:
We determine concavity by the sign of f′′:
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At a stationary point (f′(a)=0),
If f′′(a)>0, then f has a local minimum at x=a.
If f′′(a)<0, then f has a local maximum at x=a.
Using the second derivative to classify a stationary point is often called the second derivative test.
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Inflexion points occur when f′′(x)=0 and f′′(x) changes sign. 🚫
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When f′ crosses the x-axis f has a maximum (f′′<0) or minimum (f′′>0)
When f′′ crosses the x-axis, f has an inflexion point.
The nth derivative of f is
Given three variables x, y, and z,
Hence, given dzdx, we can find an expression for dzdy by calculating dxdy.
Implicit differentiation is when we differentiate both sides of an equation. It is helpful when we have an equation that cannot be simplified to y=f(x).
Since dtdy=dxdy⋅dtdx, you may be asked to use implicit differentiation to find dxdy, then with a given dtdx and point, you can find dtdy.
Given the time rate of change of radius, length, height, or width of a three dimensional object, you may find the time rate of change of volume by taking the derivative of the volume equation.
Let L be the distance from the origin of a point with coordinates (x,y). Then, given dtdx and dtdy, we can find dtdL at a given point (x,y).
Using a given rate of change dtdx and trigonometry, we can calculate dxdθ, which can be used to find dtdθ.
If g(a)f(a)=00 or ∞∞, then: