# Chapter 2 (Differentiation) FSc Part 2 Math

Level: F.Sc (Pre-Engg)
Class: 2nd Year
Board: Punjab Textbook Board
Subject: Mathematics
Total Exercises: 10
Type: Notes

Is there actually a way to “unlock math” for ourselves and learn the subject quickly? It is a universal and undebatable knowledge that mathematics is a darn difficult subject.

In this chapter we will read these topics:

• Name the mathematicians who invented different calculus.
• Define dependent and independent varaibles.
• Define the derivative of a function.
• What is differentiation?
• Differentiation by definition.
• The first principle or AB-Initio Method.
• Write different notations used for derivatives by different mathematicians.
• Find derivative when n is a positive integer.
• Find derivative when n is a negative integer.
• Find derivative when n is zero.
• Find from the definition, the differential coefficient of the given variable.
• Find from the 1st principle.
• Theorems on differentiation
• State and prove the theorem of derivatives.
• State and prove the difference theorem of the derivative.
• State and prove the product theorem of derivatives.
• State and prove the quotient theorem of derivatives.
• State and prove the reciprocal rule of the derivative.
• The Chain Rule.
• A derivative of Parametric equations.
• Define Parameter.
• Define Parametric equations.
• Define explicit and implicit functions.
• A derivative of Implicit Relations.
• Differentiation of trigonometric functions.
• Prove by the ab-initio method.
• Find derivatives from the first principle.
• Derivation of inverse trigonometric functions.
• Differentiation of exponential functions.
• Define the exponential function.
• Define general exponential function.
• Define natural exponential function.
• Differentiation of logarithmic functions.
• Define the general logarithm function.
• Define common logarithm function.
• Define natural logarithm function.
• Differentiation of hyperbolic functions.
• Define hyperbolic functions.
• A derivative of inverse hyperbolic functions.
• Higher derivatives.
• First derivative
• Second derivative
• Third derivative.
• Fourth derivative.
• Nth derivative.
• Higher derivative.
• Series expansions of functions
• Define Maclaurin’s series expansion of a function.
• State and prove Maclaurin’s series expansion of a function.
• Find Maclaurin’s series for sinx.
• State and prove Taylor’s series expansion of a function.
• Geometrical meaning of derivative.
• State the geometrical interpretation of derivative.
• Define the tangent line.
• Prove that geometrically derivative of a function gives the slope of a tangent line.
• Discuss the tangent line to the graph of function |x| at x=0.
• Define increasing function.
• Define a decreasing function.
• Define a constant function.
• Define an increasing function in terms of derivatives.
• Define decreasing function in terms of derivatives.
• Define constant function in terms of derivatives.
• Define stationary point.
• Define critical value and critical point.
• Define turning point of a function.
• Define point of inflection.
• Define relative minima of a function.
• Define relative maxima of a function.
• What is the relative extrema of a function.
• Define first derivative test.
• Applications of maxima and minima.
• Find two positives integers whose sum is 9 and the product of one with the square of the other will be maximum.
• What are dimensions of a box of a square base having largest volume if the sum of one side of the base and height is 12cm.
• The perimeter of a triangle is 20cm. If one side is of the length 8cm, what are length of the other sides for the mum area of the triangle?
• FInd the two positive integers whose sum is 30 and their product will be maximum.