Blog 4; Derivatives

Finding The Derivative

Today, we are going to discuss how to find the derivative of a couple different types of functions. First of all, lets just talk about the derivative. The derivative is the rate of change, or the slope of a tangent line, at a point on a function. In order to find that rate of change, that slope at that point, you need to either find the slope of a linear tangent line that passes through that point or find the derivative of the function itself and plug in the point as your ‘x’ value. This is important to keep in mind when thinking about our first two types functions and how to find their derivative.

First, we have the derivative of a constant function, which you can see an example of that type of graph above. A constant function is just a function with no slope so we can think about it like a linear function, mx + b, where b is the y intercept. Essentially, a constant function has a 0 for ‘m’ and anything multiplied by 0 is 0 so ‘x’ is also gone and we are left with just B. In simpler terms, a constant function is a horizontal line on a graph and because it has no slope, it similarly has no derivative. Remember finding the derivative is finding the slope at a point, so if there is no slope there is no derivative, thus the derivative of a constant function is 0.

f(x) = C, or f(x) = 7, derivative of a constant function = 0

The second type of function we are going to talk about is a linear function, you can see an example above. Remember a linear function is a function with the form y = mx + b, where ‘m’ is the slope and ‘b’ is the y-intercept. The derivative of a linear function is equal to the slope of that function because the slope is the same at all points. The slope of a linear function has a constant slope and therefore all points have the same slope and the same derivative. Therefore the derivative of y = mx + b is equal to m.

f(x) = mx + b è f’(x) = m

The third type of function we have here is a power function. You will recognize these functions as parabola’s at the squared level and as more complex graphs as the power is raised. To find the derivative of a power function, you need to use the power rule. The power rule states that the derivative of xⁿ is equal to nx^n-1 where the exponent is dropped in front and multiplied to the x and the x drops one exponent. Thus the derivative of a cubic function x³ is equal to 3x², which says the derivative at any point in the function x³ is equal to 3 times that point raised to the second power.

f(x) = xⁿ è f’(x) = nx^n-1

            The fourth type of function is actually quite similar in appearance to the power function, however the operation needed to find its derivative is quite different. A power function, as mentioned previously, is a base of ‘x’ raised to some number ‘n’ whereas an exponential function is a function where some number B is raised to a variable like ‘x.’ An example might be that a power function is x⁵ whereas a exponential function is 5^x.  To find the derivative we have to multiply that b^x by the natural log of b, so the derivative of b^x = (ln b)(b^x). There is also the interesting case of an exponential function with a base of ‘e’ which you can see a graphed example of, along with an example of log graph which we'll get to, just below this section. We know that 'e' is just a number and thus just like the other exponential functions we just looked at, we can’t use the power rule we have to find the derivative another way. It just so happens that the derivative of e^x is actually just e^x, however if that exponent ‘x’ has a constant, ‘k’, multiplied against it like e^kx , we can use something that feels like the power rule. Here we take that constant, ‘k’, and multiply it to the front of the ‘e’, but we keep that constant as is in the exponent. For example; the derivative of e^2x = 2e^2x

f(x) = b^x è f’(x) = (ln b)(b^x)

f(x) = e^x è f’(x) = e^x

f(x) = e^kx è f’(x) = ke^kx

We just talked a little about logarithmic functions when discussing exponential functions so now is good time to discuss how you find the derivative of a logarithmic function. To find the derivative of a logarithmic function, you have to look out for a couple of things. At the most basic level a log has an invisible 1 multiplied in the front and inside the log itself, for example log x is also 1*log 1x. We don’t write this because as you all know, when you multiply one against something it stays the same. However when we think about the derivative of a simple log function its important to think about those ones. The derivative of a log function is equal to the number in front of the log over the number inside of the log function. For example, if we have 3log 2x the derivative of that function is equal to 3/2x.

f(x) = klog nx è f’(x) = k/nx

f(x) = log x è f’(x) = 1/x

Next we are going to look at trigonometric functions. The two Trig functions we’ll look at are the sine, or sin, function and cosine, or cos, function. Once again we need to keep an eye on those invisible 1’s, because they’ll help you understand why the derivative looks one way in one situation and another way in different situation. Firstly, the derivative of a sin function is a cos function; the derivative of sin x is just cos x. Secondly, the derivative of a cos function is a negative sin function, or the derivative cos x is equal to -sin x. This is where those invisible 1’s come in. If we have numbers other than those ones, for example 3sin 2x, the derivative is going to be altered. We once again take that inner number and multiply it against that outer number, so 3sin 2x becomes 6cos 2x. We do not change the inner part of the cos or sin function.

f(x) = ksin nx è f’(x) = nk cos nx

f(x) = kcos nx è f’(x) = - nk sin nx

Now, you may be asking your self what happens when we have different functions all thrown in together in the same equation. Well the answer is both simple and complicated. In certain polynomial functions, where each part of the polynomial is combined by only addition or subtraction all we have to do is separate each part of the equations, separately derive them, and then add/subtract them together. If we have to multiply or divide two different parts of the polynomial then there are further rules we have to follow that we will look at next time.

Here is an example of a polynomial with only addition and subtraction è

f(x) = 3x² + 47x – 12 è f’(x) = (2*3x^{2 -1}) + 47 – 0 à this one is using the power rule in the first, the linear in the second, and the constant rule in the last part.

f(x) = 2x⁴ + 2log 4x + 9x - 2e^2x è f’(x) = (4*2x^{4 -1}) + 2/4x + 9 + (2*2e^2x) à this one is using the power rule first, it takes the derivative of a log function second, and linear function third, and finally an exponential function derivative with a base ‘e’