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- Introduction
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- In Chapter,
- we developed the concept of the derivative of a function.
- We now know that the derivative f' of a function f measures the instantaneous rate of change of f with respect to x.
- The derivative also tells us the slope of the tangent line to y=f(x) at any given value of x.
- So far, we have focused on interpreting the derivative graphically or,
- in the context of a physical setting,
- as a meaningful rate of change.
- To calculate the value of the derivative at a specific point,
- we have relied on the limit definition of the derivative,
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- f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}
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- In this chapter,
- we investigate how the limit definition of the derivative
- leads to interesting patterns and rules that enable us to find a formula for f'(x) quickly,
- without
- using the limit definition directly.
- For example,
- we would like to apply shortcuts to differentiate a function such as g(x) = 4x^7 - \sin(x) + 3e^x simply by observation.
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- Some Key Notation
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- In addition to our usual f' notation,
- there are other ways to denote the derivative of a function,
- as well as the instruction to take the derivative.
- If we are thinking about the relationship between y and x,
- we sometimes denote the derivative of y with respect to x by the symbol
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- \frac{dy}{dx}
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- which we read dee-y dee-x.
- For example, if y = x^2,
- we'll write that the derivative is \frac{dy}{dx} = 2x.
- This notation comes from the fact that the derivative is related to the slope of a line,
- and slope is measured by \frac{\Delta y}{\Delta x}.
- Note that while we read \frac{\Delta y}{\Delta x} as
- change in y over change in x,
we view
- \frac{dy}{dx} as a single symbol,
- not a quotient of two quantities.
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- We use a variant of this notation as the instruction to take the derivative.
- In particular,
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- \frac{d}{dx}\left[ \Box \right]
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- means take the derivative of the quantity in \Box with respect to x.
- For example, we may write \frac{d}{dx}[x^2] = 2x.
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- It is important to note that the independent variable can be different from x.
- If we have f(z) = z^2,
- we then write f'(z) = 2z.
- Similarly, if y = t^2,
- we say \frac{dy}{dt} = 2t.
- And it is also true that \frac{d}{dq}[q^2] = 2q.
- This notation may also be used for second derivatives:
- f''(z) = \frac{d}{dz}\left[\frac{df}{dz}\right] = \frac{d^2 f}{dz^2}.
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- In what follows,
- we'll build a repertoire of functions for which we can quickly compute the derivative.
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- Constant, Power, and Exponential Functions
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- So far, we know the derivative formula for two important classes of functions:
- constant functions and power functions.
- If
- f(x) = c is a constant function,
- its graph is a horizontal line with slope zero at every point.
- Thus, \frac{d}{dx}[c] = 0.
- We summarize this with the following rule.
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- The derivative of a constant function
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- derivativeconstant function
- For any real number c,
- if f(x) = c, then f'(x) = 0.
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- If f(x) = 7, then f'(x) = 0.
- Similarly, \frac{d}{dx} [\sqrt{3}] = 0.
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- In your work in Preview Activity,
- you conjectured that for any positive integer n,
- if f(x) = x^n, then f'(x) = nx^{n-1}.
- This rule can be formally proved for any positive integer n,
- and even for any nonzero real number
- (positive or negative).
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- The derivative of a power function
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- derivativepower function
- For any nonzero real number n,
- if f(x) = x^n, then f'(x) = nx^{n-1}.
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- Using the rule for power functions,
- we can compute the following derivatives.
- If g(z) = z^{-3}, then g'(z) = -3z^{-4}.
- Similarly, if h(t) = t^{7/5},
- then \frac{dh}{dt} = \frac{7}{5}t^{2/5},
- and \frac{d}{dq} [q^{\pi}] = \pi q^{\pi - 1}.
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- It will be helpful to have a derivative formula for one more type of basic function.
- For now, we simply state this rule without explanation or justification;
- we explore why this rule is true in one of the exercises and we will encounter graphical reasoning for why the rule is plausible in Preview Activity.
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- The derivative of an exponential function
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- derivativeexponential function
- For any positive real number a,
- if f(x) = a^x, then f'(x) = a^x \ln(a).
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- If f(x) = 2^x,
- then f'(x) = 2^x \ln(2).
- Similarly, for p(t) = 10^t,
- p'(t) = 10^t \ln(10).
- It is especially important to note that when a = e,
- where e is the base of the natural logarithm function,
- we have that
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- \frac{d}{dx} [e^x] = e^x \ln(e) = e^x
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- since \ln(e) = 1.
- This is an extremely important property of the function e^x:
- its derivative function is itself!
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- Note carefully the distinction between power functions and exponential functions:
- in power functions, the variable is in the base, as in x^2,
- while in exponential functions,
- the variable is in the power, as in 2^x.
- As we can see from the rules,
- this makes a big difference in the form of the derivative.
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- Constant Multiples and Sums of Functions
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- Next we will learn how to compute the derivative of a function constructed as an algebraic combination of basic functions.
- For instance,
- we'd like to be able to take the derivative of a polynomial function such as
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- p(t) = 3t^5 - 7t^4 + t^2 - 9
- ,
- which is a sum of constant multiples of powers of t.
- To that end, we develop two new rules:
- the Constant Multiple Rule and the Sum Rule.
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- How is the derivative of y = kf(x) related to the derivative of y = f(x)?
- Recall that when we multiply a function by a constant k,
- we vertically stretch the graph by a factor of |k|
- (and reflect the graph across y = 0 if k \lt 0).
- This vertical stretch affects the slope of the graph,
- so the slope of the function
- y = kf(x) is k times as steep as the slope of y = f(x).
- Thus, when we multiply a function by a factor of k,
- we change the value of its derivative by a factor of k as well.
- The Constant Multiple Rule can be formally proved as a consequence of properties of limits,
- using the limit definition of the derivative.
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- The Constant Multiple Rule
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- constant multiple rule
- For any real number k,
- if f(x) is a differentiable function with derivative f'(x),
- then \frac{d}{dx}[k f(x)] = k f'(x).
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- In words, this rule says that the derivative of a constant times a function is the constant times the derivative of the function.
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- If g(t) = 3 \cdot 5^t,
- we have g'(t) = 3 \cdot 5^t \ln(5).
- Similarly, \frac{d}{dz} [5z^{-2}] = 5 (-2z^{-3}).
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- Next we examine a sum of two functions.
- If we have y = f(x) and y = g(x),
- we can compute a new function
- y = (f+g)(x) by adding the outputs of the two functions:
- (f+g)(x) = f(x) + g(x).
- Not only is the value of the new function the sum of the values of the two known functions,
- but the slope of the new function is the sum of the slopes of the known functions.
- Therefore
- Like the Constant Multiple Rule,
- the Sum Rule can be formally proved as a consequence of properties of limits,
- using the limit definition of the derivative.
- ,
- we arrive at the following Sum Rule for derivatives:
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- The Sum Rule
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- sum rule
- If f(x) and g(x) are differentiable functions with derivatives f'(x) and g'(x) respectively,
- then \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x).
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- In words, the Sum Rule tells us that
- the derivative of a sum is the sum of the derivatives.
- It also tells us that a sum of two differentiable functions is also differentiable.
- Furthermore, because we can view the difference function
- y = (f-g)(x) = f(x) - g(x) as y = f(x) + (-1 \cdot g(x)),
- the Sum Rule and Constant Multiple Rules together tell us that \frac{d}{dx}[f(x) + (-1 \cdot g(x))] = f'(x) - g'(x),
- or that the derivative of a difference is the difference of the derivatives.
- We can now compute derivatives of sums and differences of elementary functions.
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- Using the sum rule, \frac{d}{dw} (2^w + w^2) = 2^w \ln(2) + 2w.
- Using both the sum and constant multiple rules,
- if h(q) = 3q^6 - 4q^{-3},
- then h'(q) = 3 (6q^5) - 4(-3q^{-4}) = 18q^5 + 12q^{-4}.
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- In the same way that we have shortcut rules to help us find derivatives,
- we introduce some language that is simpler and shorter.
- Often, rather than say take the derivative of f,
- we'll instead say simply differentiate f.
- Similarly,
- if the derivative exists at a point,
- we say f is differentiable at that point,
- or that f can be differentiated.
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- As we work with the algebraic structure of functions,
- it is important to develop a big picture view of what we are doing.
- Here, we make several general observations based on the rules we have so far.
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- The derivative of any polynomial function will be another polynomial function,
- and that the degree of the derivative is one less than the degree of the original function.
- For instance, if p(t) = 7t^5 - 4t^3 + 8t,
- p is a degree 5 polynomial, and its derivative,
- p'(t) = 35t^4 - 12t^2 + 8, is a degree 4 polynomial.
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- The derivative of any exponential function is another exponential function:
- for example, if g(z) = 7 \cdot 2^z,
- then g'(z) = 7 \cdot 2^z \ln(2),
- which is also exponential.
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- We should not lose sight of the fact that all of the meaning of the derivative that we developed in Chapter still holds.
- The derivative measures the instantaneous rate of change of the original function,
- as well as the slope of the tangent line at any selected point on its graph.
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+ Additional Practice
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+ Foundations
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+ Sometimes before computing a derivative, it is helpful to rewrite the function in a more standard form. Let's review how to write radicals as power functions, as well as powers of x in the denominator of a fraction as power functions.
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+ We start with some practice with basic rules of exponents.
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