Rules for Finding Derivatives - Whitman College
We start with the derivative of a power function, f(x) = xn. Here n is a number of any kind: integer, rational, positive, negative, even irrational, as in xπ. We have already computed some simple examples, so the formula should not be a complete surprise: d dx xn = nxn−1. It is not easy to show this is true for any n.
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456 Chapter 17 Diﬀerential Equations 17.1 First Order l Differentia tions Equa We start by considering equations in which only the ﬁrst derivative of the function appears. DEFINITION 17.1.1 A ﬁrst order diﬀerential equation is an equation of the form
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The derivative function f0(x) is sometimes also called a slope-predictor function. The following is a four-step process to compute f0(x) by de nition. Input: a function f(x) Step 1 Write f(x+ h) and f(x). Step 2 Compute f(x + h) f(x). Combine like terms. If h is a common factor of the
22 Derivative of inverse function 22.1 Statement Any time we have a function f, it makes sense to form is inverse function f 1 (although this often requires a reduction in the domain of fin order to make it injective). If we know the derivative of f, then we can nd the derivative of f 1 as follows: Derivative of inverse function. If fis a ...
The derivative of an exponential function would be determined by the use of the chain rule, which was covered in the previous section. In reviewing the derivative rules for exponential functions we will begin by looking at the derivative of a function with the constant raised to a simple variable. Derivative of an exponential function in the ...
function then we need to find the derivative of the cost function. When the marginal function is evaluated it will give the approximate change for the next unit. For example, if we evaluated a marginal cost function when x = 100 then the value of C′(100) would be the approximate cost of producing the next unit (or the 101st unit).
The ﬁrst derivative of the function f(x), which we write as f0(x) or as df dx, is the slope of the tangent line to the function at the point x. To put this in non-graphical terms, the ﬁrst derivative tells us how whether a function is increasing or decreasing, and by how much it is increasing or decreasing. This information is
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What happens if we take the derivative of a generating function? As an exam-ple, let’s differentiate the now-familiar generating function for an inﬁnite sequence of 1’s: 1CxCx2Cx3Cx4CD 1 1 x IMPLIES d dx.1CxCx2Cx3Cx4C /D d dx 1 1 x IMPLIES 1C2xC3x2C4x3CD 1.1 x/2 IMPLIES h1;2;3;4;:::i ! 1.1 x/2: (12.1)
is a function and the function together with its derivatives appear in the equation. Example Given a constant k ∈ R, ﬁnd all solutions f : R → R to the diﬀerential equation f 0(x) = k f (x). Solution: Multiply the equation above f 0(x) − kf (x) = 0 by e−kx, that is, f 0(x) e−kx − f (x) ke−kx = 0. The left-hand side is a total ...