Differentiating Exponentials When the Base is Not e
Ifthenis a very familiar result. If howeverorthen differentiating is not so easy. We have to change the base first toand then differentiate. We do this using the relationship
The last expression is of the formwhich differentiates toApplying this example towe obtainDifferentiatingonly introduces another factorto giveWe can then find tangents and normals in the usual way.
Example: Find the equation of the tangent and normal to the curveat the point with coordinates
For the tangent
so at the point
For the normal