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Implicit Differentiation

Implicit Differentiation. By Samuel Chukwuemeka (Mr. C). Implicit Functions and Differentiation. In Calculus, Y is known as the dependent variable X is known as the independent variable Implicit functions are functions in which: y is expressed implicitly with x

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Implicit Differentiation

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  1. Implicit Differentiation By Samuel Chukwuemeka (Mr. C)

  2. Implicit Functions and Differentiation • In Calculus, • Y is known as the dependent variable • X is known as the independent variable • Implicit functions are functions in which: • y is expressed implicitly with x • y is not expressed in terms of x; rather the function contains the sum, difference, product and/or quotient of both variables together • Implicit Differentiation is: • the differentiation of implicit functions • the rate of change of y with respect to (wrt) changes in x of implicit functions

  3. What do we need to know before we can differentiate implicitly? • Rules of Differentiation • Power Rule: if y = axn then dy/dx = naxn – 1 • Sum Rule: if y = u + v where y, u and v are functions of x then dy/dx = du/dx + dv/dx

  4. Rules of Differentiation • Product Rule: if y = u * v • Where u and v are functions of x, • then dy/dx = u* (dv/dx) + v * (du/dx) • Function of a Function Rule or Chain Rule: • If y is a function of u and u is a function of x, • If y = f(u) and u = f(x), • Then dy/dx = dy/du * du/dx

  5. Rules of Differentiation • Quotient Rule: if y = u/v • Where y, u and v are functions of x, • Then dy/dx = (v * (du/dx) - u * (dv/dx)) / v2

  6. What else do I need to know? • Besides these rules, it is also important to know that: • Differentiating: • y wrt x dy/dx • y2 wrt x 2ydy/dx • y3 wrt x 3y2dy/dx and so on……………

  7. Can we do some examples? • Differentiate implicitly x3y + y3x = -7 • Solution • What rules do we need: Power, Product and Sum Rules • Differentiating gives that • 3x2y + x3dy/dx + y3 + 3xy2dy/dx = 0 • x3dy/dx + 3xy2dy/dx = 0 – 3x2y – y3 • dy/dx (x3 + 3xy2) = -3x2y – y3 • dy/dx = - (3x2y + y3) / (x3 + 3xy2) • Please ask your questions. • Ok. Go to the “Test” section and attempt the questions.

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