Shawn M. answered • 04/10/21
Encouraging and knowledgeable science and math tutor
Since d/dx (ln [x]) = (1/x) dx you will need to use the chain rule to solve for f'(x) and then substitute that x=0 to get the value of the derivative of the function, at this x value
So we will have with the polynomial function in the ln ( ): the coefficient of 7 stays the same when you take the derivative of the function, as does any coefficient (whether an integer, decimal value, etc. so d/dx [8x] = 8), and then you will have the 2nd order polynomial inverted, and then multiply that by the derivative of the 2nd polynomial, which will have two terms (d/dx of any constant like 1 in the 2nd order polynomial has a value of 0, since its value does not depend on the value of x), and so then at that point you can have a ratio of two polynomials as your f'(x), then lastly substitute the value of x=0. When all this is simplified, you should have this expression: -7*(-3/1) which gives an answer of 21. Where some students might go off course is first substituting in that x = 0 into the f(x) and then trying to take the derivative of that which is just a constant, which gives 0 as the answer. Or they might forget to use the chain rule, which would give an answer of =7*1 or -7.
