Second Holiday Reflection

for this blog im going to review how to solve for derivatives the hard way. Two things you need to know if you’re trying to solve for a derivative and your looking at a graph is a secant line and a tangent line. A secant line is a line that touches a graph in one spot. A tangent line touches the graph in two spots. When trying to solve for a derivative you use the formula f(x+∆x)-f(x)/∆x to help you solve it. When the directions say y', f'(x), dy/dx, d/dx, Dx[y], slope of a tangent line, or slope of the curve at a point it is telling you to find the derivative. Let’s try an example:
Find dy/dx of x²+1 at the points (0,1) & (-1,2)
=(x+∆x)²+1-(x²+1)/∆x
=x²+2x∆x+∆x²+1-x²-1/∆x
=2x∆x+∆x²/∆x
=∆x(2x+∆x)/∆x
=2x+∆x
=2x
Then you take the two points and plug it into the equation
Slope at (0, 1) = 2(0) = 0
Slope at (-1, 2) = 2(-1) = -2