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