Calculus/Differentiation/Exercises

From Wikibooks, the open-content textbooks collection

Jump to: navigation, search
← Implicit differentiation Calculus Extrema and Points of Inflection →
Differentiation/Exercises

Contents

[edit] Find The Derivative By Definition

Find the derivative of the following functions using the definition of the derivative:

  1. y = x^2\,
  2. y = 2x + 2\,
  3. y = \frac{1}{2}x^2\,
  4. y = 2x^2 + 4x + 4\,

Solutions

[edit] Prove Differentiation Rules

Use the definition of the derivative to prove the following rules:

1. For any fixed real number c, \frac{d}{dx}\left[c\right]=0.

2. For any fixed real numbers m and c, \frac{d}{dx}\left[mx+c\right]=m

3. For any fixed real number c, \frac{d}{dx}\left[cf(x)\right] = c \frac{d}{dx}\left[f(x)\right]

4. \frac{d}{dx}\left[f(x)\pm g(x)\right]= \frac{d}{dx}\left[f(x)\right]\pm\frac{d}{dx}\left[g(x)\right]

Solutions

[edit] Find The Derivative By Rules

Find the derivative of the following functions:

  1. 2x^2 + 4\,
  2. 3\sqrt[3]{x}\,
  3. \frac{1}{x^2}+3x^\frac{1}{3}\,
  4. \ln x - 2e^x + \sqrt{x}\,
  5. \sin(x)+\cos(x)\,
  6. (x+5)^2\,
  7. \sqrt{1+x^2}\,
  8. x^3 * 2\sqrt{y}\,
  9. 2\sqrt{x}*\frac{1}{y^4}\,
  10. \sqrt{2x^2+1}*(3y^4+2y)^2\,
  11. 4^x\,
  12. 2^{x-3}*3\sqrt{x^3-2}+\ln x\,
  13. xe^x*y^2*z^4\,
  14. \log_4 x + 2\ln x\,
  15. 3e^x-4\cos (x) - \frac{1}{4}\ln x\,
  16. 2x5 + 8x2 + x − 78
  17. 7x7 + 8x5 + x3 + x2 − x

Solutions

[edit] Implicit Differentiation

  1. x2 + y2 = 1
  2. x3 + y3 = xy

Solutions

[edit] Higher Order Derivatives

  1. What is the second derivative of 3x4 + 3x2 + 2x?

Solutions

[edit] Set One A: ?

Note: There are currently no answers given for these exercises.

Solutions to Set One A

1. Find whether the following functions are increasing or decreasing in interval of (2,3)

i) 10-6x-2x2 ii) 2x3-12x2+18x+15 iii) 5+36x+3x2-2x3

iv) 8+36x+3x2-2x3 v) 5x3-15x2-120x+3 vi) x3-6x2-36x+2

[edit] Set Two

Note: There are currently no answers given for these exercises.

Q1: Show that minimum value of -->

    x2-4x+9   is 5.

Q2: Show that :

    the expression x+ 1/x  cannot have any value intermediate b/w 2 and -2.

Q3: Show that :

    (x2+x+1)/(x2-x+1)  has 3 for its maximum value  , 
    and 1/3 for its minimum value.


Q4: Show that :

    the value of (x2+px+1)/(x2-px+1) is intermediate
    b/w (2+p)/(2-p) and (2-p)/(2+p).


Q5: Show that :

    (ax2+2bx+c)/(ax2+2bx+a)  is unlimited value if 
    a+c<2b.


Q6: Show that :

    the shortest distance from a given point to a given st. line is the 
    perpendicular distance.


Q7: Show that the greatest triangle inscribed in a given circle is equilateral.

Navigation: Main Page · Precalculus · Limits · Differentiation · Integration · Parametric and Polar Equations · Sequences and Series · Vector Calculations · Multivariable Calculus & Differential Equations · Extensions · References

Retrieved from "http://en.wikibooks.org/wiki/Calculus/Differentiation/Exercises"