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Derivatives of Logarithmic Functions - Problem 1

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

One important derivative to memorize is that the derivative of ln(x) is 1/x. This can be used when differentiating functions that involve the natural log. When given a function that involves both a polynomial and a natural log, you can differentiate as you normally would by using the sum rule (the derivative of a sum is equal to the sum of derivatives). So, differentiate the polynomial as you normally would. Then, recall that the derivative of the natural log is 1/x, and derive accordingly. For example, if f(x)=4x2-3+5ln(x), the derivative of 4x2 is 8x, the derivative of -3 is 0, and the derivative of 5ln(x) is 5*(1/x). So, f'(x)=8x+5/x.

So we just learned that the derivative of a natural log lnx is 1 over x. I just wanted to show you a picture once again of the function of f(x) equals natural log, and its derivative function. Remember that the y coordinate of every point on this curve is going to be the natural log of the x coordinate. But it's slope.

The slope at every point is going to be given by this curve in red. So for example, at x equals 1. Natural log of 1 is 0, and at that point, the slope is going to be 1. So every point in this curve gives you the slope of this curve. Let's use our derivative of natural log to differentiate this function; x² plus 4x minus 3lnx.

So this is going to be dy/dx that we're looking for. That's the derivative with respect to x of x² plus 4x minus 3lnx. So we should use our properties of derivatives here. We have the sum property and the constant multiple rule. I can break the sum up into the derivative of x² plus the derivative of 4x plus the derivative of -3lnx. I can pull this constant of -3 out. So this is going to be plus -3 times the derivative of lnx.

Now, the derivative of 4x, that's just a linear function. Derivative of a linear function is the slope, so 4. The derivative of x² is 2x. So all we have to do is take the derivative of this piece, and of course that's 1 over x. So we have 2x plus 4 plus -3 times 1 over x. That's 2x plus 4 minus 3 over x. That's the derivative of y equals x² plus 4x minus 3lnx.

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