Differentiating Exponential Functions Differentiating Exponential Functions A function is that mathematical term which states relationship between constants and one or more variables. For example, consider a function f(x) = 7x4 + 100, which expresses a relationship between the variable ‘x’ that is raised to some power 4 and constants 7 and 100. We are aware of the word differentiation and its use. It can also be called as derivative and is represented as dy / dx or f'(x). While differentiating exponential Functions we need to follow a formula that is given as follows: d (ax) /d x = x ax-1. The same rule is applied for finding the further Derivatives also. To understand it better let us consider an example of a function that is given as:: F (v) = 10 v7 – 41 v6 + 4 v4 + 15 v2 + 10 v To differentiate this function we will follow the above mentioned formula and the general rules of differentiation. So, we can write the derivative with respect to the variable v as:
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F’ (v) = d (10 v7 – 41 v6 + 4 v4 + 15 v2 + 10 v) / dv, differentiating each term in the function with respect to v: F’ (v) = d (10 v7) / dv – d (41 v6) / dv + d (4 v4) / dv+ d (15 v2) / dv+ d (10 v) / dv As the differentiation of constant values results to zero we are left with: F’ (v) = 10 d (v7) / dv – 41 d (v6) / dv + 4 d (v4) / dv+ 15 d (v2) / dv+ 10 d (v) / dv Using the formula we get: F’ (v) = (7 * 10 (v7-1)) – (6 * 41 (v6-1)) + (4 * 4 (v4-1)) + (15 * 2 (v2-1)) + (10 *1 (v1-1)) Or F’ (v) = 70 v6 – 246 v5 + 16 v3 + 30 v1 + 10 (First derivative) In the similar way the other higher order derivatives can be finding easily. The exponential function ex is perhaps the easiest function to differentiate: it is the only function whose derivative is the same as the function itself. In a nutshell, d/dx(ex)=ex. Slightly more complicated is e2x or e3x and so on. In these cases the number in front of the x comes out to the the front in the derivative, as with the trig functions. So d/dx(e2x)=2e2x, d/dx(e3x)=3e3x and so on. More generally, d/dx(eax)=aeax. If the number that's being raised to the power x is not "e" but some other number, then it's a little more complicated to differentiate the function. Cinsider the function 2x for example. To differentiate that with respect to x, we need to take logs first, to bring the x down. We'll give the function 2x a name, let's call it y, so y(x)=2x.
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Now we take logs. This gives us ln(y(x))=ln(2x)=xln(2), using a log law to bring the x to the front. Next we differentiate that equation with respect to x, to get 1/y(x)*dy/dx=ln(2). Notice that we differentiated the ln(y(x)) as a nested function, the inner function being y(x). Remember that we're looking for dy/dx, the derivative of 2x. We can get this by rearranging the result we just obtained, so: dy/dx=ln(2)*y(x) Substituting for y(x) we find dy/dx=ln(2)*2x, in other words d/dx(2x)=ln(2)*2x. The same process works for 3x, or 4x and so on, in fact we can write down the general rule: d/dx (ax)=ln(a)*ax for any value of a.
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