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IBMathsDPAnalysis & Approaches (AA)HLFlashcards5. CalculusDifferentiation

Differentiation (DP IB Analysis & Approaches (AA))

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  • What is a limit?

    A limit is a value that a function approaches as x approaches a particular value from either side.

  • What is a derivative?

    A derivative (also known as a gradient function) is a function that relates the gradient of another function to the value of x.

  • What does fraction numerator straight d y over denominator straight d x end fraction mean?

    fraction numerator straight d y over denominator straight d x end fraction means the derivative of y with respect to x.

  • What does f to the power of apostrophe stretchy left parenthesis x stretchy right parenthesis mean?

    f to the power of apostrophe open parentheses x close parentheses means the derivative of the function f open parentheses x close parentheses with respect to x.

  • True or False?

    The derivative of a constant function is always zero.

    True.

    The derivative of a constant function is always zero.

  • State the formula for differentiating x to the power of n.

    If f open parentheses x close parentheses equals x to the power of n, then f to the power of apostrophe open parentheses x close parentheses equals n x to the power of n minus 1 end exponent.

    I.e. the original power comes in front as a multiplier, then the original power is reduced by 1.

    This formula is in the exam formula booklet.

  • What is the derivative of a x, where a is a constant?

    The derivative of a x, where a is a constant, is a.

  • What is a tangent?

    A tangent is a straight line that touches a curve at a point without crossing through it.

  • True or False?

    The gradient of a curve at a point is equal to the gradient of the tangent to the curve at that point.

    True.

    The gradient of a curve at a point is equal to the gradient of the tangent to the curve at that point.

  • What is an alternative notation for f to the power of apostrophe open parentheses x close parentheses?

    If y equals f open parentheses x close parentheses, then an alternative notation for f to the power of apostrophe open parentheses x close parentheses is fraction numerator straight d y over denominator straight d x end fraction.

  • True or False?

    The derivative of 4 over x is 4 x to the power of negative 2 end exponent.

    False.

    The derivative of 4 over x is negative 4 x to the power of negative 2 end exponent, which is the same as negative 4 over x squared.

    To differentiate a reciprocal, first rewrite it as a negative power: 4 over x equals 4 x to the power of negative 1 end exponent.

    Then use "If f open parentheses x close parentheses equals x to the power of n, then f to the power of apostrophe open parentheses x close parentheses equals n x to the power of n minus 1 end exponent".

  • What is the derivative of a sum (or difference) of functions equal to?

    The derivative of a sum (or difference) of functions is equal to the sum (or difference) of their individual derivatives.

  • True or False?

    The formula for differentiating powers of bold italic x, that says if f open parentheses x close parentheses equals x to the power of n then f to the power of apostrophe open parentheses x close parentheses equals n x to the power of n minus 1 end exponent, can only be used when the power n is an integer.

    False.

    The formula for differentiating powers of bold italic x, that says if f open parentheses x close parentheses equals x to the power of n then f to the power of apostrophe open parentheses x close parentheses equals n x to the power of n minus 1 end exponent, can be used when the power n is any rational number.

  • What does it mean if f to the power of apostrophe open parentheses x close parentheses greater than 0?

    If f to the power of apostrophe open parentheses x close parentheses greater than 0, it means the function f open parentheses x close parentheses is increasing.

  • True or False?

    A function is decreasing if f to the power of apostrophe open parentheses x close parentheses less than 0.

    True.

    A function is decreasing if f to the power of apostrophe open parentheses x close parentheses less than 0.

  • What is a normal (or normal line) with regard to a function?

    A normal (or normal line) is a straight line that passes through a point on a curve and is perpendicular to the tangent at that point.

  • What is the relationship between the gradients of a tangent and a normal at the same point on a curve?

    At the same point on a curve, the product of the gradients of a tangent and its normal is equal to -1.

  • True or False.

    The equation of the tangent to y equals f open parentheses x close parentheses at the point open parentheses x subscript 1 comma space y subscript 1 close parentheses can be found by using y minus y subscript 1 equals f open parentheses x subscript 1 close parentheses open parentheses x minus x subscript 1 close parentheses.

    False.

    To find the equation of the tangent you need to use the value of the derivative (gradient function) of the curve at the point open parentheses x subscript 1 comma space y subscript 1 close parentheses.

    The equation of the tangent to y equals f open parentheses x close parentheses at the point open parentheses x subscript 1 comma space y subscript 1 close parentheses can be found by using y minus y subscript 1 equals f to the power of apostrophe open parentheses x subscript 1 close parentheses open parentheses x minus x subscript 1 close parentheses.

  • What is an equation for the normal to a curve y equals f open parentheses x close parentheses at the point open parentheses x subscript 1 comma space y subscript 1 close parentheses?

    An equation for the normal to a curve y equals f open parentheses x close parentheses at the point open parentheses x subscript 1 comma space y subscript 1 close parentheses is y minus y subscript 1 equals negative fraction numerator 1 over denominator f to the power of apostrophe open parentheses x subscript 1 close parentheses end fraction open parentheses x minus x subscript 1 close parentheses.

Previous:Normal DistributionNext:Techniques & Applications of Differentiation

    1. Topic 1

      Number & Algebra Toolkit

    2. Topic 2

      Exponentials & Logs

    3. Topic 3

      Sequences & Series

    4. Topic 4

      Simple Proof & Reasoning

    5. Topic 5

      Proof by Induction & Contradiction

    6. Topic 6

      Binomial Theorem

    7. Topic 7

      Permutations & Combinations

    8. Topic 8

      Complex Numbers

    9. Topic 9

      Further Complex Numbers

    10. Topic 10

      Systems of Linear Equations

    1. Topic 1

      Linear Functions & Graphs

    2. Topic 2

      Quadratic Functions & Graphs

    3. Topic 3

      Functions Toolkit

    4. Topic 4

      Other Functions & Graphs

    5. Topic 5

      Reciprocal & Rational Functions

    6. Topic 6

      Transformations of Graphs

    7. Topic 7

      Polynomial Functions

    8. Topic 8

      Inequalities

    9. Topic 9

      Modulus Functions & Further Transformations

    1. Topic 1

      Geometry Toolkit

    2. Topic 2

      Geometry of 3D Shapes

    3. Topic 3

      Trigonometry

    4. Topic 4

      The Unit Circle & Exact Values

    5. Topic 5

      Trigonometric Functions & Graphs

    6. Topic 6

      Trigonometric Equations & Identities

    7. Topic 7

      Inverse & Reciprocal Trigonometric Functions

    8. Topic 8

      Trigonometric Proof & Equation Strategies

    9. Topic 9

      Vector Properties

    10. Topic 10

      Vector Equations of Lines

    11. Topic 11

      Vector Planes

    1. Topic 1

      Statistics Toolkit

    2. Topic 2

      Correlation & Regression

    3. Topic 3

      Probability

    4. Topic 4

      Discrete Random Variables

    5. Topic 5

      Binomial Distribution

    6. Topic 6

      Normal Distribution

    7. Topic 7

      Continuous Random Variables

    1. Topic 1

      Differentiation

    2. Topic 2

      Techniques & Applications of Differentiation

    3. Topic 3

      Integration

    4. Topic 4

      Techniques & Applications of Integration

    5. Topic 5

      Optimisation

    6. Topic 6

      Kinematics

    7. Topic 7

      Basic Limits & Continuity

    8. Topic 8

      Further Differentiation

    9. Topic 9

      Further Integration

    10. Topic 10

      Differential Equations

    11. Topic 11

      Maclaurin Series

    12. Topic 12

      Limits using l'Hôpital's Rule & Maclaurin Series