the natural log function
The calculator shows two graphs, the first one is the exponential graph we did in the last section. The second graph is the natural log graph (y = ln x). The two graphs are reflections of each other in the y = x axis (the line that runs 45 degrees sloping upwards left to right- I'll sketch that in later).
You should notice that when x = 2, the gradient of the graph y = ln x is 0.5 this means the gradient (differentiated function) = 1/x since 1/2 = 0.5.
Another way of stating this is to take the reciprocal of everything inside the bracket then differentiate everything inside the bracket (two steps). So we need some examples using the rule below. Don't forget why we are doing this - differentiate a function so as to find the gradient or slope or rate at any point along the path of the function. If you think this is challenging think about Newton and Leibnitz trying to educate the great intellectual minds of their day with this - it wasn't easy! For example, the astronomer Halley needed Newton to help him calculate the paths of orbits of bodies travelling around the sun so as to predict the return of his "Halley's Comet" in 1758. Newton's calculus made this a reality. Halley, Hooke and Wren used to meet regularly in coffee houses to discuss scientific issues. Halley knew he would never live to see his comet return - but sure enough , 16 years after he died it showed up on time as he predicted.
You should notice that when x = 2, the gradient of the graph y = ln x is 0.5 this means the gradient (differentiated function) = 1/x since 1/2 = 0.5.
Another way of stating this is to take the reciprocal of everything inside the bracket then differentiate everything inside the bracket (two steps). So we need some examples using the rule below. Don't forget why we are doing this - differentiate a function so as to find the gradient or slope or rate at any point along the path of the function. If you think this is challenging think about Newton and Leibnitz trying to educate the great intellectual minds of their day with this - it wasn't easy! For example, the astronomer Halley needed Newton to help him calculate the paths of orbits of bodies travelling around the sun so as to predict the return of his "Halley's Comet" in 1758. Newton's calculus made this a reality. Halley, Hooke and Wren used to meet regularly in coffee houses to discuss scientific issues. Halley knew he would never live to see his comet return - but sure enough , 16 years after he died it showed up on time as he predicted.