Blog #3 - Mary Graci

This is the third Calculus blog of the year. Last week, after learning about taking a derivative, we learned a shortcut. It’s really quite simple, easy to remember, and SO much faster than doing it the long way. Here are the rules:
1. the constant rule: the derivative of a constant is always 0
Ex. 1) d/dx 7 = 0

2. the power rule: whenever you take the derivative of something with an exponent, you lose a power. The formula for this shortcut is d/dx (x^n) = nx^(n-1)
Ex. 2) d/dx 3x^2
First, bring the exponent to the front (and if there is a constant in front of the x, multiply the two). Then, subtract one from the exponent.
= 2(3)x^(2-1) = 6x
Ex. 3) d/dx (x^3 + 9x)
When there are multiple terms in an equation, take the derivative of each term individually.
= 3x^2 + 9

We also learned how to tell if the equation is not differentiable (meaning you can’t take a derivative.) First of all, you it’s not differentiable if there is a corner on the graph. This happens with absolute values, and some piecewise.
Ex. 4) absolute value of (x-3)
There is an absolute value, therefore this equation is not differentiable at x = 3.

Second, you can’t take the derivative of a vertical tangent line (meaning when there is an x raised to an odd # over another odd #).
Ex. 5) (x+5)^3/5
Because of the exponent 3/5, this equation is not differentiable at x = -5.

Third, you can’t take the derivative if it is not continuous (such as with a jump, vertical asymptote, or removable).

And lastly, you can’t take the derivative of a cusp (meaning when there is an x raised to an even # over an odd #).
Ex. 6) x^(6/7)
Because of the exponent 6/7, this equation is not differentiable at x = 0.

I pretty much understood everything this week, especially with the shortcut making everything so much easier!