A standard calculus problem which actually involves very little calculus is logarithmic differentiation. The reason logarithmic differentiation is effective are logarithm rules that transform products to sums AND exponents to factors. Also the rule for taking the derivative of the logarithm of a function is easy.
Many math professors really pine about how much easier logarithmic differentiation is, but it is about the same amount of work as just using the product rule combined with the chain rule and then some slightly difficult algebra at the end. The big difference between logarithmic differentiation and the method I just mentioned is that you do NOT have to do any factoring in logarithmic differentiation ONLY some simplification involving exponents at the end of the problem.
The following problem was solved in 6 steps.
- Step 1 involved utilizing the log of a product becomes a sum AND exponents become factors.
- Step 2 was the only calculus step. In this step we utilized the rule stating the derivative of the logarithm of a function is the derivative of the function divided by the function.
- Step 3 combined two fractions into a single fraction having a common denominator. Note the similarity to this and cross multiplying two equal fractions.
- Step 4 Multiplied and collected like terms
- Step 5 and 6 are a little tricky. Since we are trying to find y prime (derivative), we must multiply both sides of equality by y. When one does this, the exponents 5 and 4 become 4 and 3 respectively.
- If you have any criticisms or need any clarification, please let me know. If you liked it, you may donate a quarter via the PayPal donation button just below the comment window. Thank you. And yes, that is some dirty blue carpet that my wife asked me to vacuum today. Better get that done!