2.3 Basic Differentiation Formulas
In this section we get to use some shortcuts to take derivatives of constant functions, power functions, polynomials, and the sine and cosine functions.
First consider the constant function f (x) = c. Next we consider the power functions f (x) = xn, where n is a positive integer.
f (x) = x
f (x) = x2
f (x) = x3
Ex: f (x) = x8
f (x) = x100
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Now consider a power functions f (x) = xn, where n is not a positive integer.
f (x) = x ­1
f (x) = x1/2
Ex: f (x) = x ­7
f(x)=∛x
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New Derivatives from Old:
When new functions are formed from old functions by addition, subtraction, or multiplication by a constant, their derivatives can be calculated in terms of the derivatives of the old functions as follows: Ex: f(x) = 5x2
Using prime notation, we get (f + g)' = f ' + g' Ex: f(x) = 5x2 + 7x + 10
Using prime notation, we get (f ­ g)' = f ' ­ g' Ex: f(x) = x5 ­ x­3 ­ √x
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The points on the graph of a function where the tangent line is horizontal occur when the derivative is zero. Use this information to find the points on the graph of f(x) below where the tangent line is horizontal.
The line which is perpendicular to the tangent line is called the normal line. Find the equations of the tangent line and the normal line to the graph of the function above at the point (1, ­2).
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Find the slopes of the tangent and normal lines to the graph of the function f(x) = √x
at the point (9, 3).
Find the equations of the tangent line and the normal line to the graph of the function below at the point (3, 22).
f(x) = 5x2 ­ 10x + 7
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Differentiate each function below using the methods of this section.
f(x) = (5x + 3)(2x ­ 7)
y = (4x ­ 7)2
f(x) = 62 ­ π3 + 4000 + eπ
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The Sine and Cosine Functions: If we sketch the graph of y = sin x and use it to graph y', we see that it looks suspiciously like y = cos x. It turns out that, in fact, i.e. If f(x) = sin x, then f '(x) = cos x.
Now sketch the graph of y = cos x and use it to graph y'.
Find y' for each of the given functions.
y = 5cos x + 7
y = x2 + 4sin x
y = f(θ)= 6sinθ + 8cosθ
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Find the equations of the tangent and normal lines to the graph of the function f(x) = 5cos x
at the point (π/2, 0).
Find the equations of the tangent and normal lines to the graph of the function f(x) = sin x ­ 10x
at the point (π, ­10π).
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Recall that the velocity function is the derivative of the position function (Δposition/Δtime). Similarly, the acceleration function is the derivative of the velocity function (Δvelocity/Δtime). Use the position function f (t) = t3 ­ 12t, where t is in seconds and f(t) is in meters, to do the following.
a) Find the velocity at time t.
b) Find the velocity at time 3 seconds.
c) When is the particle at rest?
d) When is the particle moving in the positive direction?
In the negative direction?
e) Find the total distance traveled by the particle in the first 4 seconds.
f) Find the acceleration at time t.
g) Find the acceleration at time 3 seconds.
h) Graph the position, velocity, and acceleration functions for time 0 ­ 4 seconds.
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