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Finding Derivatives

Finding Derivatives. Sections 2.1, 2.2, 5.1, and 5.4. Definition. The derivative of a function tells us the instantaneous rate of change of a function. We can see the derivative by looking at the average rate of change over a decreasing interval. The definition of derivative.

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Finding Derivatives

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  1. Finding Derivatives Sections 2.1, 2.2, 5.1, and 5.4

  2. Definition • The derivative of a function tells us the instantaneous rate of change of a function. • We can see the derivative by looking at the average rate of change over a decreasing interval.

  3. The definition of derivative • The derivative of x2 + x would be given by

  4. Use the definition of definition of derivative to find the derivative of each of the following functions: • x2 + 5x - 7 • 3x2 – 4x + 6

  5. The real way to take a derivative • For a polynomial, we can take the derivative in two steps. • 1. Bring down the exponent and multiply it times the coefficient of x. • 2. Subtract 1 from each exponent of x. • Find the derivative of 3x2 – 5x + 6

  6. Find the derivative for each of the following • x2 + 6x – 7 • 3x2 + 5x + 2

  7. More Derivatives • Before you try to take the derivative, make sure that everything is converted to a rational exponent.

  8. ex and ln x • The derivative of ex is ex • The derivative of ln x is 1/x • Find the derivative of 3x2 + 5x – x-1/2+ ln x

  9. Finding slope • Since the derivative tells the rate of change of a function, we can substitute a specific x-value into the derivative to find the slope of the tangent line to the curve at a point. • Find the slope of the tangent line to each curve at x = 4. • x2 + 3x – 2 • 2x2 + 5 – 1/x

  10. Equation of Tangent Line • Find the equation of a tangent line at x = 4 for each of the following functions. • -3x2+ 17 • 2x2 + 5 –

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