2.2 Definition Of Derivativeap Calculus

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Access Single Variable Essential Calculus 2nd Edition Chapter 2.2 Problem 20E solution now. Our solutions are written by Chegg experts so you can be assured of the highest quality! For y = 1/(2x + 3) use the limit definition of derivative to calculate y. For y = 1/(2 x + 3) use the limit definition of derivative to calculate y’. For notes and practice problems, visit the Calculus course on Calculus (Version #1) is created for a 45-minute class period and f.
The derivative of a function is one of the basic concepts of mathematics. Together with the integral, derivative occupies a central place in calculus. The process of finding the derivative is called differentiation. The inverse operation for differentiation is called integration.
The derivative of a function at some point characterizes the rate of change of the function at this point. We can estimate the rate of change by calculating the ratio of change of the function (Delta y) to the change of the independent variable (Delta x). In the definition of derivative, this ratio is considered in the limit as (Delta x to 0.) Let us turn to a more rigorous formulation.Formal Definition of the Derivative
*The derivative of a function is the measure of change in that function. Consider the parabola y=x^2. For negative x-values, on the left of the y-axis, the parabola is decreasing (falling down towards y=0), while for positive x-values, on the right of the y-axis, the parabola is increasing (shooting up from y=0).
*Derivative (calculus) synonyms, Derivative (calculus) pronunciation, Derivative (calculus) translation, English dictionary definition of Derivative (calculus). Resulting from or employing derivation: a derivative word; a derivative process.
Let (fleft( x right)) be a function whose domain contains an open interval about some point ({x_0}). Then the function (fleft( x right)) is said to be differentiable at ({x_0}), and the derivative of (fleft( x right)) at ({x_0}) is given by
[
{f’left( {{x_0}} right) = limlimits_{Delta x to 0} frac{{Delta y}}{{Delta x}} }
= {limlimits_{Delta x to 0} frac{{fleft( {{x_0} + Delta x} right) – fleft( {{x_0}} right)}}{{Delta x}}.}
]
Lagrange’s notation is to write the derivative of the function (y = fleft( x right)) as (f^primeleft( x right)) or (y^primeleft( x right).)
Leibniz’s notation is to write the derivative of the function (y = fleft( x right)) as (large{frac{{df}}{{dx}}}normalsize) or (large{frac{{dy}}{{dx}}}normalsize.)
The steps to find the derivative of a function (fleft( x right)) at the point ({x_0}) are as follows:
* Form the difference quotient ({largefrac{{Delta y}}{{Delta x}}normalsize} = {largefrac{{fleft( {{x_0} + Delta x} right) – fleft( {{x_0}} right)}}{{Delta x}}normalsize});
* Simplify the quotient, canceling (Delta x) if possible;
* Find the derivative (f’left( {{x_0}} right)), applying the limit to the quotient. If this limit exists, then we say that the function (fleft( x right)) is differentiable at ({x_0}).
In the examples below, we derive the derivatives of the basic elementary functions using the formal definition of derivative. These functions comprise the backbone in the sense that the derivatives of other functions can be derived from them using the basic differentiation rules.Solved Problems
Click or tap a problem to see the solution.Example 1Using the definition of derivative, prove that the derivative of a constant is (0.)Example 2Calculate the derivative of the function (y = x.)Example 3Using the limit definition find the derivative of the function (fleft( x right) = 3x + 2.)Example 4Find the derivative of a linear function (y = ax + b) using the definition of derivative.Example 5Using the definition, find the derivative of the simplest quadratic function (y = {x^2}.)Example 6Using the definition of the derivative, differentiate the function (fleft( x right) = {x^2} + 2x – 2.)Example 7Determine the derivative of a quadratic function of general form (y = a{x^2} + bx +c.)Example 8Using the definition of the derivative, find the derivative of the function (y = largefrac{1}{x}normalsize.)Example 9Using the limit definition find the derivative of the function (fleft( x right) = frac{1}{{x – 1}}.)2.2 Definition Of Derivativeap Calculus 14th EditionExample 10Find the derivative of the function (fleft( x right) = large{frac{2}{{{x^2}}}}normalsize) using the limit definition.Example 11Find the derivative of the function (y = sqrt x .)Example 12Determine the derivative of the cube root function (fleft( x right) = sqrt[3]{x}) using the limit definition.Example 13Calculate the derivative of the cubic function (y = {x^3}.)Example 14Differentiate the power function (fleft( x right) = {x^4}) using the limit definition.Example 15Find the derivative of the sine function (y = sin x.)Example 16Find the derivative of the cosine function (y = cos x.)Example 17Find the derivative of the trigonometric function (fleft( x right) = sin 2x) using the limit definition.Example 18Find an expression for the derivative of the exponential function (y = {e^x}) using the definition of derivative.Example 19Find the derivative of the power function (y = {x^n}.)Example 20Find the derivative of the natural logarithm (y = ln x.)Example 1.Using the definition of derivative, prove that the derivative of a constant is (0.)
Solution.
In this case, the function (yleft( x right)) is always equal to to a constant (C.) Therefore, we can write
[{yleft( x right) = C,;;;}kern-0.3pt {yleft( {x + Delta x} right) = C.}]
It is clear that the increment of the function is identically equal to zero:
[
{Delta y = yleft( {x + Delta x} right) – yleft( x right) }
= {C – C equiv 0.}
]
Substituting this in the limit definition of derivative, we obtain:
[
{y’left( x right) = limlimits_{Delta x to 0} frac{{Delta y}}{{Delta x}} }
= {limlimits_{Delta x to 0} frac{{yleft( {x + Delta x} right) – yleft( {x} right)}}{{Delta x}} }
= {limlimits_{Delta x to 0} frac{0}{{Delta x}} }
= {limlimits_{Delta x to 0} 0 = 0.}
]Example 2.Calculate the derivative of the function (y = x.)
Solution.
Following the above procedure, we form the ratio (largefrac{{Delta y}}{{Delta x}}normalsize) and find the limit as (Delta x to 0:)
[require{cancel}
{y’left( x right) = limlimits_{Delta x to 0} frac{{Delta y}}{{Delta x}} }
= {limlimits_{Delta x to 0} frac{{left( {x + Delta x} right) – x}}{{Delta x}} }
= {limlimits_{Delta x to 0} frac{{cancel{x} + Delta x – cancel{x}}}{{Delta x}} }
= {limlimits_{Delta x to 0} frac{{cancel{Delta x}}}{{cancel{Delta x}}} }
= {limlimits_{Delta x to 0} 1 = 1.}
]Example 3.Using the limit definition find the derivative of the function (fleft( x right) = 3x + 2.)
Solution.
Write the increment of the function:
[require{cancel}{Delta y }={ yleft( {x + Delta x} right) – yleft( x right) }={ left[ {3left( {x + Delta x} right) + 2} right] – left[ {3x + 2} right] }={ cancel{color{blue}{3x}} + 3Delta x + cancel{color{red}{2}} – cancel{color{blue}{3x}} – cancel{color{red}{2}} }={ 3Delta x.}]2.2 Definition Of Derivativeap Calculus Solver
The difference ratio is equal to
[frac{{Delta y}}{{Delta x}} = frac{{3cancel{Delta x}}}{{cancel{Delta x}}} = 3.]
Then the derivative is given by
[{f^primeleft( x right) = mathop {lim }limits_{Delta x to 0} frac{{Delta y}}{{Delta x}} }={ mathop {lim }limits_{Delta x to 0} 3 }={ 3.}]Example 4.Find the derivative of a linear function (y = ax + b) using the definition of derivative.
Solution.
We write the increment of the function corresponding to a small change in the argument (Delta x:)
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[
{Delta y = yleft( {x + Delta x} right) – yleft( x right) }
= {left( {aleft( {x + Delta x} right) + b} right) – left( {ax + b} right) }
= {cancel{color{blue}{ax}} + aDelta x + cancel{color{red}{b}} – cancel{color{blue}{ax}} – cancel{color{red}{b}} = aDelta x.}
]
Then the derivative is given by
[
{y’left( x right) = limlimits_{Delta x to 0} frac{{Delta y}}{{Delta x}} }
= {limlimits_{Delta x to 0} frac{{acancel{Delta x}}}{{cancel{Delta x}}} }
= {limlimits_{Delta x to 0} a = a.}
]
As it can be seen, the derivative of a linear function (y = ax + b) is always constant and equal to the coefficient (a.)Example 5.Using the definition, find the derivative of the simplest quadratic function (y = {x^2}.)
Solution.
If we change the independent variable (x) by an amount (Delta x), the function receives the following increment:
[
{Delta y = yleft( {x + Delta x} right) – yleft( x right) }
= {{left( {x + Delta x} right)^2} – {x^2}.}
]2.2 Definition Of Derivativeap Calculus Algebra
This expression can be converted to the form
[
{Delta y = {left( {x + Delta x} right)^2} – {x^2} }
= {cancel{x^2} + 2xDelta x + {left( {Delta x} right)^2} – cancel{x^2} }
= {left( {2x + Delta x} right)Delta x.}
]
By calculating the limit, we find the derivative:
[
{y’left( x right) }
= {limlimits_{Delta x to 0} frac{{Delta y}}{{Delta x}} }
= {limlimits_{Delta x to 0} frac{{left( {2x + Delta x} right)cancel{Delta x}}}{{cancel{Delta x}}} }
= {limlimits_{Delta x to 0} left( {2x + Delta x} right) = 2x.}
]Example 6.Using the definition of the derivative, differentiate the function (fleft( x right) = {x^2} + 2x – 2.)
Solution.
Calculate the increment of the function:
[require{cancel}{Delta y }={ fleft( {x + Delta x} right) – fleft( x right) }={ left[ {{{left( {x + Delta x} right)}^2} + 2left( {x + Delta x} right) – 2} right] }-{ left[ {{x^2} + 2x – 2} right] }={ cancel{color{darkgreen}{x^2}} + 2xDelta x + {left( {Delta x} right)^2} + cancel{color{blue}{2x}} + 2Delta x – cancel{color{red}{2}}} – {cancel{color{darkgreen}{x^2}} – cancel{color{blue}{2x}} + cancel{color{red}{2}} }={ left( {2x + 2} right)Delta x + {left( {Delta x} right)^2}.}]
Write the difference ratio:
[{frac{{Delta y}}{{Delta x}} }={ frac{{left( {2x + 2} right)Delta x + {{left( {Delta x} right)}^2}}}{{Delta x}} }={ frac{{left( {2x + 2 + Delta x} right)cancel{Delta x}}}{{cancel{Delta x}}} }={ 2x + 2 + Delta x.}]
Hence, the derivative is
[{f^primeleft( x right) }={ mathop {lim }limits_{Delta x to 0} frac{{Delta y}}{{Delta x}} }={ mathop {lim }limits_{Delta x to 0} left( {2x + 2 + Delta x} right) }={ 2x + 2.}]Example 7.Determine the derivative of a quadratic function of general form (y = a{x^2} + bx +c.)
Solution.
We find the derivative of the given function using the definition of derivative. Write the increment of the function (Delta y) when the argument changes by (Delta x:)
[require{cancel}
{Delta y = yleft( {x + Delta x} right) – yleft( x right) }
= {left[ {a{{left( {x + Delta x} right)}^2} + bleft( {x + Delta x} right) + c} right] }-{ left[ {a{x^2} + bx + c} right] }
= {cancel{color{blue}{a{x^2}}} + 2axDelta x + a{left( {Delta x} right)^2} }+{ cancel{color{red}{bx}} + bDelta x + cancel{color{maroon}{c}} }-{ cancel{color{blue}{a{x^2}}} – cancel{color{red}{bx}} – cancel{color{maroon}{c}} }
= {2axDelta x + a{left( {Delta x} right)^2} + bDelta x }
= {left( {2ax + b + aDelta x} right)Delta x.}
]
Now we form the ratio of the increments and calculate the limit:
[
{y’left( x right) }
= {limlimits_{Delta x to 0} frac{{Delta y}}{{Delta x}} }
= {limlimits_{Delta x to 0} frac{{left( {2ax + b + aDelta x} right)cancel{Delta x}}}{{cancel{Delta x}}} }
= {limlimits_{Delta x to 0} left( {2ax + b + aDelta x} right) }
= {2ax + b.}
]
Thus, the derivative of a quadratic function in general form is a linear function.Problems 1-72.2 Definition Of Derivativeap Calculus CalculatorProblems 8-20
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