Chapter 8. Equation: Calculus: Differentiation CALCULUS
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1 Chapter 8 Calculus: Differentiation CALCULUS Introduction Slope of linear equation Slope of a curve Calculus Differential calculus Integral calculus The dependent variable is said to be a function of the independent variable and the equation describes this link. In terms of y and x, y is said to be a function of x (written as y = f(x)). y = f(x) = a + bx The slope of a straight line does not change along the length of the line i.e. it is the same at every point on the line. The slope of a curve changes along the length of the line. Calculus is the mathematical study of change. Differential calculus (differentiation). This is concerned with the rate of change of a function (represented by the slope of a curve). Integral calculus (integration). This is concerned with change in areaor volume. QUADRATIC EQUATIONS Quadratic equations take the form The equation and its curve Line of symmetry The line of a quadratic equation is either shaped like a u (when the coefficient of x 2 is positive) or like an inverted u (when the coefficient of x 2 is negative). The curve is called a parabola. A parabola is symmetrical about a central line (called the line of symmetry). A parabola has a turning point (vertex) where it turns from positive to negative or negative to positive. The line of symmetry is a plane that runs through the centre of the quadratic curve. The curve on one side of the line of symmetry is the mirror image of the curve on the other side. The line of symmetry is always a vertical line of the form x = n. It passes throughthe turning point. The value of x axis that the line of symmetry passes through can be found using the following expression. Turning point (vertex) The turning point lies on the line of symmetry. The y coordinate can be found by substituting the value of x for the line of symmetry back into the quadratic equation or by using a formula. Estimating the slope (Derivative) Differentiating sums (Sum Rule) Equation: If an equation is comprised of a series of terms added to or subtracted from each other, each term is differentiated separately. The derivative of the equation is the sum of the derivatives for each term (sum rule). Page 1 of 19 (kashifadeel.com)
2 QUESTION BANK BASIC AND GENERAL 01 (c) (d) None of these ANSWER 01 To avoid quotient rule, we removed the reciprocal form Derivative using basic rule (c) 30 (d) 20 ANSWER 02 Simple Average of change (not a differential calculus question) Page 2 of 19 (kashifadeel.com)
3 QUESTION BANK SUM 03 The differential coefficients of is (c) (d) None of these ANSWER 03 To avoid quotient rule, we removed the reciprocal form Derivative using Sum Rule Alternatively, Derivative using Quotient rule 04 The gradient of the curve is: (c) 0 (d) None of above ANSWER 04 Note: Gradient = slope & Slope = First derivative Sum Rule At x=0 05 (c) (d) None of these ANSWER 05 It is better that we multiply all the factors Sum Rule Page 3 of 19 (kashifadeel.com)
4 06 (c) (d) None of these ANSWER 06 Sum Rule Evaluate the options [ ] 07, the expression is equal to 0-1 (c) 1 (d) None of these ANSWER 07 [ ] Sum Rule [ ] Given that Solved Page 4 of 19 (kashifadeel.com)
5 08 (c) (d) ANSWER 08 Sum Rule Page 5 of 19 (kashifadeel.com)
6 DIFFERENTIATION OF MORE COMPLEX FUNCTIONS Differentiating products (Product Rule) Differentiating Quotients (Division Rule) Where u and v are factors of the product. Where u is the dividend and v is the divisor. Differentiating function of a function (Basic Rule) Differentiating function of a function (Chain rule) Derivative of y with respect to q Derivative of q with respect to x It may be necessary to use chain rule within questions requiring differentiation of a product or of a quotient. Page 6 of 19 (kashifadeel.com)
7 QUESTION BANK QUOTIENT RULE 09 (c) (d) ANSWER 09 Quotient Rule 10 is: (c) (d) None of these ANSWER 10 Quotient Rule Page 7 of 19 (kashifadeel.com)
8 11 (c) (d) None of these ANSWER 11 Quotient Rule 12 (c) (d) None of these ANSWER 12 side so we have to separate r on left hand Quotient Rule ( ) Page 8 of 19 (kashifadeel.com)
9 13 is equal to (c) (d) None of these ANSWER 13 Quotient Rule ( ) Page 9 of 19 (kashifadeel.com)
10 14 For the following function, the second derivative [ ] at x=3 is: (c) (d) ANSWER 14 Quotient Rule with Second derivative Quotient Rule Again [ ] Page 10 of 19 (kashifadeel.com)
11 QUESTION BANK FUNCTION OF FUNCTION 15 The derivative of (c) ANSWER 15 (d) None of these Function of Function 16 (c) 0 (d) None of these ANSWER 16 Function of Function ( ( ( ( ) ) [ ] ) [ ] ) ) Page 11 of 19 (kashifadeel.com)
12 QUESTION BANK CHAIN RULE 17 t (c) (d) None of these ANSWER 17, Chain Rule Question Basic Rule Basic Rule Reciprocal of both sides Chain rule Page 12 of 19 (kashifadeel.com)
13 QUESTION BANK MIXED AND COMPLEX 18 (c) (d) ANSWER 18 Chain Rule = Quotient Rule Sum Rule Chain Rule = X=1 Page 13 of 19 (kashifadeel.com)
14 19 (c) 2 (d) None of these ANSWER 19 ( ) Quotient Rule Multiplying both sides by denominator Divided by 2 Quadratic Alternatively, calculate and match each option in turn with solution of quadratic figure. Page 14 of 19 (kashifadeel.com)
15 DIFFERENTIATION OF SPECIAL RELATIONSHIP Equation Derivative Exponential Functions Log functions The above rule apply for natural logs. For log of a different base, the base can be changed to log natural to facilitate differentiation. Page 15 of 19 (kashifadeel.com)
16 QUESTION BANK EXPONENTIAL RULE 20 (c) (d) None of these ANSWER 20 Exponential Rule Page 16 of 19 (kashifadeel.com)
17 QUESTION BANK MIXED 21 (c) (d) None of these ANSWER 21 Exponential Rule and Function of Function Rule 22 : (c) (d) None of these ANSWER 22 Quotient Rule and Exponential Rule Page 17 of 19 (kashifadeel.com)
18 23 (c) e (d) None of these ANSWER 23 Quotient Rule and Exponential Rule [ ] Given that x=1 24 (c) (d) ANSWER 24 Product and exponential Rule Multiplicative identity Page 18 of 19 (kashifadeel.com)
19 25 is: (c) (d) ANSWER 25 [ ] Exponential, Logarithmic, Product Rule Product rule Product rule again and log rule Credits: Mr. Waseem Saeed and Ms. Anum Iqbal Concept & Compilation: Kashif Adeel Dated: 02 December 2017 Page 19 of 19 (kashifadeel.com)
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