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Derivatives of Exponential Functions
Derivative of ex y= eu y1= eu · du
Examples of  Derivative of ex Differentiate f (x)=e4x²−6x+9 y1 = e4x²−6x+9 · du  du = 8x-6  y1 = (8x-6)e4x²−6x+9
Examples of  Derivative of ex y1= cosxesinxy= esinx y1= esinx · du  Find the du of sinx Remember Trig. Derivatives sinx= cosx y1= esinx · cosx
Second Example y=esinxtanx y1=esinxtanx· du  du = sinxtanx ( Product Rule) du = sinxsec²x+ tanxcosx y1=(sinxsec²x+ tanxcosx)esinxtanx
Third Example  y = e5x + x² y = e5x + x² · du  du = 5x + x² du= 5 +2x ( Simple Power Rule )  y = (5+2x)e5x + x²
Remember!!  The derivative of exis quite remarkable. The expression for the derivative is the same as the expression that we started with; that is, ex! This means that the slope of a tangent line to the curve y = ex at any point is equal to the y-coordinate of the point.

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Derivatives Of Exponential Functions

  • 2. Derivative of ex y= eu y1= eu · du
  • 3. Examples of Derivative of ex Differentiate f (x)=e4x²−6x+9 y1 = e4x²−6x+9 · du du = 8x-6 y1 = (8x-6)e4x²−6x+9
  • 4. Examples of Derivative of ex y1= cosxesinxy= esinx y1= esinx · du Find the du of sinx Remember Trig. Derivatives sinx= cosx y1= esinx · cosx
  • 5. Second Example y=esinxtanx y1=esinxtanx· du du = sinxtanx ( Product Rule) du = sinxsec²x+ tanxcosx y1=(sinxsec²x+ tanxcosx)esinxtanx
  • 6. Third Example y = e5x + x² y = e5x + x² · du du = 5x + x² du= 5 +2x ( Simple Power Rule ) y = (5+2x)e5x + x²
  • 7. Remember!! The derivative of exis quite remarkable. The expression for the derivative is the same as the expression that we started with; that is, ex! This means that the slope of a tangent line to the curve y = ex at any point is equal to the y-coordinate of the point.