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Derivatives of Exponential Functions and Tangent Lines

Learn how to calculate the derivatives of 2x, bx, and ex, and find equations of tangent lines to exponential curves. Includes step-by-step examples and solutions.

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Derivatives of Exponential Functions and Tangent Lines

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  1. §4.2 The Exponential Function ex

  2. Section Outline • e • The Derivatives of 2x, bx, and ex

  3. The Number e

  4. The Derivative of 2x

  5. Solving Exponential Equations EXAMPLE Calculate. SOLUTION

  6. The Derivatives of bx and ex

  7. Solving Exponential Equations EXAMPLE Find the equation of the tangent line to the curve at (0, 1). SOLUTION We must first find the derivative function and then find the value of the derivative at (0, 1). Then we can use the point-slope form of a line to find the desired tangent line equation. This is the given function. Differentiate. Use the quotient rule.

  8. Solving Exponential Equations CONTINUED Simplify. Factor. Simplify the numerator. Now we evaluate the derivative at x = 0.

  9. Solving Exponential Equations CONTINUED Now we know a point on the tangent line, (0, 1), and the slope of that line, -1. We will now use the point-slope form of a line to determine the equation of the desired tangent line. This is the point-slope form of a line. (x1, y1) = (0, 1) and m = -1. Simplify.

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