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Mastering Differentiation for Polynomial, Trigonometric, Exponential, & Logarithmic Functions

Learn to differentiate polynomial, trigonometric, exponential, and logarithmic functions, solve problems involving composite functions, products, and quotients, and find higher derivatives efficiently.

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Mastering Differentiation for Polynomial, Trigonometric, Exponential, & Logarithmic Functions

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  1. Differentiation – Learning Outcomes • Differentiate polynomial, trigonometric, exponential, and logarithmic functions. • Differentiate sums and real multiples of these functions. • Solve problems about composite functions using the chain rule. • Solve problems about products and quotients using product and quotient rules. • Find higher derivatives of functions, particularly second derivatives.

  2. Differentiate Polynomial Functions • For a function , it is useful to know not just its value, but also how its value changes with . • For linear functions, , we have already studied slope, which provides this. • For more complicated functions, we use differentiation. • Differentiation is an operation which returns the derivative, , of a function – this describes how is changing around .

  3. Differentiate Polynomial Functions In words, “multiply by the current power, then reduce the power by 1” • For a simple polynomial function, , the power rule states that its derivative, . • e.g. • e.g. • This is also true for negative and fractional powers: • e.g. • e.g.

  4. Differentiate Polynomial Functions • e.g. Find the derivative of each of the following functions:

  5. Differentiate Polynomial Functions • For more complicated polynomial functions, we need two additional rules: • i.e. coefficients of a function remain in product with the derivative of the function. • e.g. • i.e. terms of a function may be differentiated independently of each other. • e.g.

  6. Differentiate Polynomial Functions • e.g. Find the derivative of each of the following functions:

  7. Differentiate Trigonometric Functions • Three trigonometric functions must be differentiated on our course:

  8. Differentiate Exponential Functions • We are only concerned with one type of exponential function in differentiation: • The rules for coefficients and sums / differences apply to exponentials as well: • e.g. • e.g.

  9. Differentiate Logarithmic Functions • We are only concerned with one type of logarithmic function in differentiation: • The rules for coefficients and sums / differences apply to logarithms as well: • e.g. • e.g.

  10. Differentiate Functions • Differentiation sometimes uses different notation, particularly in co-ordinate geometry and other areas where curves are not written as functions. • e.g. • is pronounced “the derivative of with respect to ” or just “”

  11. Solve Problems about Composite Functions • Recall composite functions from chapter 1. • Previously, they were written or . • In differentiation, we typically ignore this notation unless needed. • e.g. we could write as and , then . • Instead, we ditch the function notation: • Let and . Remember your order of operations here!

  12. Solve Problems about Composite Functions • To differentiate composite functions, each function must be differentiated separately, then multiplied together. Notice how the fractions reduce! This is called the chain rule. • e.g.

  13. Solve Problems about Composite Functions • e.g. Differentiate each of the following functions:

  14. Solve Problems about Composite Functions 2003 HL P1 Q6 • Differentiate with respect to . 2004 HL P1 Q6 • Differentiate with respect to . 2005 HL P1 Q6 • Differentiate with respect to : 2012 HL P1 Q6 • Differentiate with respect to : 2011 HL P1 Q6 • The equation of a curve is • Find

  15. Solve Problems about Products • For products of functions, we need the product rule: • e.g.

  16. Solve Problems about Products • e.g. Find the derivative of each of the following:

  17. Solve Problems about Products QB T6P1 Q39 • Let . Show that: QB T6P2 Q21 • Let . Show that: 2004 OL P1 Q7 • Differentiate with respect to . 2005 OL P1 Q7 • Differentiate with respect to . 2007 OL P1 Q7 • Differentiate with respect to .

  18. Solve Problems about Quotients • For quotients of functions, we need the quotient rule: • e.g.

  19. Solve Problems about Quotients • e.g. Find the derivative of each of the following:

  20. Solve Problems about Quotients QB T6P1 Q4 • Let , for . Use the quotient rule to show that . QB T6P3 Q3 • Let , for . • Find . • Let , for . • Show that . QB T6P4 Q10 • Let • Use the quotient rule to show that

  21. Find Higher Derivatives • When a derivative is differentiated, the result is the second derivative of the original function. • Higher derivatives also exist, but are less frequently used. • Using function notation, these are written , , , , … • Using geometry notation, written , , , … • e.g. • e.g.

  22. Find Higher Derivatives • e.g. Find the second derivative of each of the following: • e.g. Find the fourth derivative of the following:

  23. Find Higher Derivatives QB T6P2 Q24 • Given that , answer the following: • Find the first four derivatives of . • Write an expression for in terms of and . 2012 HL P1 Q6 • Let . Show that the second derivative of with respect to is .

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