Derivative


Application of Derivative

  1. To find the slope of curve or to find the slope of tangent to a curve
    सामान्यतयाः

    gradient =\( \frac{\text{change in y}}{\text{change in x}} \)

    भन्ने बुझिन्छ। जसलाई तलको चित्रबाट हेरौ।
    Fig.1

    गणितमा slope ले सीधा रेखाको ठाडोपन (वा तेस्रोपन वा बाङ्गोपन) को मापन गर्दछ। सीधा रेखाको slope गणना गर्न, हामी यसमा कुनै पनि दुई बिन्दुहरू लिन्छौ र पहिलो बिन्दुबाट दोस्रो बिन्दुमा run र rise को गणना गर्छौ। Run भनेको x-निर्देशांकहरूमा हुने परिवर्तन हो र rise भनेको y-निर्देशांकहरूमा भएको परिवर्तन हो, जसलाई तलको चित्र [चित्र 1] मा चित्रण गरिएको छ।

    त्यसैले,
    slope =\( \frac{rise}{run} \) (१)
    सीधा रेखाको slope धनात्मक, ऋणात्मक वा शून्य हुन सक्छ, रजुन कुरा पहिलो बिन्दुबाट दोस्रो बिन्दुमा coordinate हरु को मान बढ्छ, घट्छ वा उस्तै रहन्छ भन्नेमा निर्भर गर्दछ।
    खास गरी, बायाँबाट दायाँ जादा तल रेखाको स्थिति तल गएमा ऋणात्मक ग्रेडियन्ट (slope) हुन्छ, तेर्सो भइमा शून्य ग्रेडियन्ट (slope) हुन्छ, र माथि गएमा धनात्मक ग्रेडियन्ट (slope) हुन्छ ।
    यदि रन (Run) को मान शून्य छ भने रेखाको ग्रेडियन्ट (slope) अपरिभाषित हुन्छ, किनकि शून्यले भाग गर्ने कुरा सम्भव छैन। त्यसैले ठाडो रेखाहरूको ग्रेडिएन्टहरू अपरिभाषित हुन्छन्।
    गणितिय रुपमा, दुईवटा बिन्दुहरू \( (x_1, y_1)\) र \( (x_2, y_2)\) छन भने ति बिन्दुहरुले बनाउने सिधा रेखाको ग्रेडियन्ट (slope) तलको चित्रमा चित्रण गरे जस्तै (चित्र 2) हुन्छ।

    Fig.2

    Then
    run = x2 − x1 and rise = y2 − y1.
    Now,
    gradient =\( \frac{y_2-y_1}{x_2-x_1} \) (2)
    Remember that when we use this formula to calculate the gradient of a straight line, it doesn’t matter which point we take to be the first point, \( (x_1, y_1)\), and which you take to be the second point, \( (x_2, y_2)\), as we get the same result either way.
    For example, using the formula to calculate the gradient of the line through the points (1, 8) and (5, 2), which is shown in Figure below [Figure 3]

    Fig.3

    It gives either
    gradient =\( \frac{y_2-y_1}{x_2-x_1}=\frac{2-8}{5-1}=-\frac{3}{2} \)
    gradient =\( \frac{y_2-y_1}{x_2-x_1}=\frac{8-2}{1-5}=-\frac{3}{2} \)
    Now,
    We define derivative as slope of tangent. Please remember, the word ‘tangent’ comes from the Latin word ‘tangere’, which means ‘to touch’. The English word ‘tangible’, which means ‘capable of being touched’, comes from the same Latin word.

    Derivative as Slope of tangent
    Fig.4

    Let y = f (x) be a differentiable function at x.
    Let P(x, f(x)) and Q(x + h, f(x+h)) be two nearby points.
    Then PQ is a secant.
    Now, the slope of the secant PQ is given by
    slope of secant at \( P = \frac{f(x+h)-f(x) }{x + h- x}\)
    or slope of secant at \( P = \frac{ f(x+h)-f(x) }{h}\) (A)
    This expression (A) is known as the difference quotient for the function f at x
    Taking limit as \( h \to 0\) in (A), the secant will turn into tangent at P ,
    Then, the slope of tangent at P is
    slope of tangent at \( P = \displaystyle \lim_{h \to 0} \frac{f(x+h)-f(x)}{ h}\)
    or slope of tangent at P = f'(x)

  2. Increasing/decreasing test of a function
    Increasing Function

    A function \( y = f (x)\) is said to be increasing in an open interval \( (a, b)\) i.e. between the points \( x = a\) and \( x = b\) if
    \( \frac{dy}{dx} \) or \( f ' (x)\) is positive
    for all values of \( x\) in that interval.
    In other words, f(x) is said to be increasing if
    \( x_1 < x_2 \Rightarrow f (x_1) < f (x_2)\)
    or equivalently
    \( x_1 > x_2 \Rightarrow f (x_1) > f (x_2)\)
    for all real numbers \( x_1\) and \( x_2\) in \( (a, b)\) .

    An increasing function can be shown graphically as in the figure below.

    The graph of such a function is a curve which goes on rising with the rise in the value of \( x\).

    Decreasing Function

    A function \( y = f (x)\) is said to be decreasing in an open interval \( (a, b)\) i.e. between the points \( x = a\) and \( x = b\) if
    \( \frac{dy}{dx} \) or \( f ' (x)\) is negative
    for all values of \( x\) in that interval.
    In other words, f(x) is said to be increasing if
    \( x_1 < x_2 \Rightarrow f (x_1) > f (x_2)\)
    or equivalently
    \( x_1 > x_2 \Rightarrow f (x_1) < f (x_2)\)
    for all real numbers \( x_1\) and \( x_2\) in \( (a, b)\) .

    A decreasing function can be shown graphically as in the figure below.

    The graph of such a function is a curve which goes on falling with the raise in the value of \( x\).

    Example 1

    Show that the function \( f(x) = 2x^3 - 3x^2 - 36x \) is increasing on each of the intervals (-∞,-2) and (3,∞), and decreasing on the interval (-2, 3).

    Solution
    Find the derivative. The derivative is
    \( f'(x) = 6x^2 - 6x - 36\) (1)
    Factorizing (i) gives
    \( f'(x) = 6(x^2 - x - 6) = 6(x + 2)(x - 3)\)

    (-∞,-2) (-2,3) (3,∞)
    (x+2) - + +
    (x-3) - - +
    f'(x) + - +
    curve increasing decreasing increasing

    1. When x is less than -2, the values of x + 2 and x - 3 are both negative, and hence the value of \( f'(x) = 6(x + 2)(x - 3) \) is positive.
      Therefore, by the increasing/decreasing criterion, the function f is increasing on the interval (-∞,-2).
    2. When x is in the interval (-2, 3), the value of x + 2 is positive and the value of x - 3 is negative, and hence the value of \( f'(x) = 6(x + 2)(x - 3)\) is negative.
      Therefore, by the increasing/decreasing criterion, the function f is decreasing on the interval (-2, 3).
    3. When x is greater than 3, the values of x + 2 and x - 3 are both positive, and hence the value of \( f'(x) = 6(x + 2)(x - 3)\) is also positive.
      Therefore, by the increasing/decreasing criterion, the function f is increasing on the interval (3,∞).
    First Derivative Test
    The value of first derivative
  3. Nature of points
    Critical Point

    Critical points are places where the derivative of a function is either zero or undefined. that is, either the graph of function has a horizontal tangent line or function is not differentiable at the point.
    In the figure below, A,B,C, and D are Critical points in which

    1. f'(x)=0 at A
    2. f'(x)=0 at B
    3. f'(x) is NOT defined at C (f is not differentiable at C)
    4. f'(x)=0 at D
    Stationary Point

    Stationary points are places where the derivative of a function is zero (only zero), that is, the graph of function has a horizontal tangent line at the point.
    In the figure below, A,B,and D are stationary points in which

    1. f'(x)=0 at A
    2. f'(x)=0 at B
    3. f'(x)=0 at D
    Note: all stationary points are critical points, but not all critical points are stationary points.
    Turning Point

    A stationary point where the derivative of the function changes sign (from positive to negative, or vice versa) at that point. Thus, a stationary point in which a function has a local maximum or local minimum is called a turning point.
    There are two types of turning points.

    1. If f is increasing on the left interval and decreasing on the right interval, then the stationary point is a local maximum
    2. If f is decreasing on the left interval and increasing on the right interval, then the stationary point is a local minimum

    In the figure below, C is only the turning point[Please not that A and B is not Turning point] in which

    1. f'(x)=0 at A, f''(x)=0 at A
    2. f'(x) is NOt defined at B B
    3. f'(x)=0 at C, local minima is defined at C
    Note: all turning points are stationary points, but not all stationary points are turning points.
    Straight line

    Some stationary points are neither turning points nor horizontal points of inflection. For example, every point on the graph of the equation \(y = 1\) (see Figure below), or on any horizontal line, is a stationary point that is neither a turning point nor a horizontal point of inflection. It is a straight line.

    Point of Inflection

    If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa. A point where this occurs is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection.

    The value of second derivative

    A point on a function f (x) is said to be a point of inflexion if f''(x) = 0 and f'''(x)≠0.
    At this point, the concavity changes from upward to downward or vice-versa.
    Therefore the point of inflexion is the transition between concavity of the curves.

    Saddle point (Horizontal point of inflection)

    A saddle point or minimax point is a point on the graph of a function where the slopes (derivatives) is zero , but which is not a local extremum of the function. Thus, a saddle point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum.
    Figure below shows the graph of the function \( f(x) = x^3\) .
    The derivative of this function is \( f'(x) = 3x^2\), so the gradient of the graph when x = 0 is
    \(f'(0) = 3 \times 0^2 = 0\).
    So the function \( f(x) = x^3\) has a stationary point at x = 0.
    However, this stationary point isn’t a turning point.
    Because
    The second derivative of this function is \( f''(x) = 6x\), so the second derivative of the graph when x = 0 is
    \(f''(0) = 6 \times 0 = 0\).
    So the function \( f(x) = x^3\) has a inflexional point at x = 0.
    In such case, the point is called saddle point

  4. Concavity of a function

    In calculus, the second derivative of a function f is the derivative of f''. Roughly speaking, the second derivative measures instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time.

    On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way.

    1. The graph of \( y = f (x)\) is concave up if \( f'' > 0\)
    2. The graph of \( y = f (x)\) is concave down if \( f'' < 0\)

    The second derivative of a function f can be used to determine the concavity of the graph of f. A function whose second derivative is positive will be concave up (also referred to as convex), meaning that the tangent line will lie below the graph of the function. Similarly, a function whose second derivative is negative will be concave down (also simply called concave), and its tangent lines will lie above the graph of the function.

    Second derivative test

    The relation between the second derivative and the graph can be used to test whether a stationary point for a function (i.e., a point where \( f'(x)=0\) is a local maximum or a local minimum. Specifically,

    1. If \( f''< 0\), then f has a local maximum at x.
    2. If \( f''> 0\), then f has a local minimum at x.
    3. If \( f''=0\), then the second derivative test says nothing about the point x, a possible inflection point.
  5. Maxima and Minima of a function
  6. Rolls' Theorem
  7. Mean Value Theorem



Derivative Ex 16.3 [BCB page 434]

  1. Find, from the first principle, the derivative of:
    1. \( \log (ax+b)\)
    2. \( \log_5 x\)
    3. \( \log \frac{x}{10} \)
    4. \( e^{ax+b} \)
    5. \( e^{\frac{x}{3}} \)
  2. Find the derivative of:
    1. \( \log (/sin x)\)
    2. \( \log (x+/tan x)\)
    3. \( \log (1+e^{5x})\)
    4. \( \log (\log x)\)
    5. \( \log (\sec x)\)
    6. \( \log (1+\sin^2 x)\)
    7. \( \ln (e^{ax}+e^{-ax})\)
    8. \( \log (\sqrt{a^2+x^2}+b)\)
    9. \( \log (\sqrt{a+x}+(\sqrt{a-x})\)
    10. \( \ln |x-4|\)
  3. Find the differential coefficient of:
    1. \( e^{\sin x} \)
    2. \( e^{\sqrt{\cos x}} \)
    3. \( e^{1+\log x} \)
    4. \( e^{\sin (\log x)} \)
    5. \( \tan (\log x) \)
    6. \( \sin (1+e^{ax}) \)
    7. \( \cos (\log \sec x) \)
    8. \( \sec (\log \tan x) \)
    9. \( \sin \log \sin e^{x^2} \)
  4. Differentiate the following with respect to x:
    1. \( x^2 \log (1+x)\)
    2. \( x^5 e^{ax}\)
    3. \( \sin ax \log x\)
    4. \( e^{ax} \cos bx\)
    5. \( (\tan x+x^2)\log x\)
    6. \( \sin x+\cos x) e^{ax} \)
  5. Calculate the derivative of:
    1. \( \frac{\log x }{\sin x} \)
    2. \( \frac{\log (ax+b) }{e^{px}} \)
    3. \( \frac{e^{ax} }{\cos bx} \)
    4. \( \frac{\sin ax }{1+\log x} \)
    5. \( \frac{\log x }{a^2+x^2 } \)
  6. Find \(\frac{dy}{dx}\) when
    1. \( xy=\log (x^2+y^2) \)
    2. \( x^2+y^2=\log (x+y)\)
    3. \( e^{xy}=xy \)
    4. \( x=e^{\cos 2t}, y=e^{\sin 2t} \)
    5. \( x=\cos (\log t), y=\log(\cos t) \)
    6. \( x=\log t+\sin t, y=e^t+\cos t \)



Practical Question

A function \(y=f(x)\) is considered, say \(y=\), now answer to the following questions.
  1. Draw the graph of the function \(y=f(x)\)
  2. Given the point on the curve \(y=f(x)\), draw a tangent at the point making an angle \(\theta\) with the positive direction of x-axis
  3. Find \(\tan \theta\)
  4. Find the point on the curve where the tangent is parallel to the x-axis, if possible
  5. Find the point on the curve where the tangent is perpendicular to the x-axis, if possible



Additional Question

  1. Define differential cofficient of a function at a given point. Find from first principles, the derivative of \( x |\sqrt{x}\) with respect to x
  2. Find the derivative of
    1. \( x^{\sin x} \)

      Solution 👉 Click Here

    2. \( (\sin x)^x \)

      Solution 👉 Click Here

    3. \( (\sin x)^{\log x} \)

      Solution 👉 Click Here

    4. \( e^{x^x} \)

      Solution 👉 Click Here

    5. \( x^{e^x} \)
    6. \( (\log x)^{\tan x} \)
    7. \( (\tan x)^{\log x} \)
    8. \( x^{\sec x} \)
    9. \( (\sin x)^{\cos x} \)
  3. Find \(\frac{dy}{dx}\) when
    1. \( y=x^y \)
    2. \( x^y.y^x=1\)
    3. \( x^m.y^n=(x+y)^{m+n}\)
    4. \( e^{\sin x}+e^{\sin y}=1\)
    5. \(x^y=y^x\)
    6. \(x^{\sin x}=y^{\sin y}\)

No comments:

Post a Comment