Differentiation
Differentiation is a function process in mathematics where the variable of interest is mapped to another value. This other value, which can be represented by \(y\), is dependent on the input value, x. The relationship between the input and output values is different for every differentiable function and always holds that:
\( y = f (x) \).
This statement defines how we will solve differentiation problems. First, we must know what kind of function we are dealing with; then we can apply knowledge about solving specific functions to derive an equation to find the derivative. It shouldn’t come as a surprise that there are many types of derivatives, but this course will focus primarily on finding derivatives of polynomials and rational functions.
Polynomial Functions
A polynomial function has the form:
\( f (x) = a_n x^n + a_{n – 1} x^{n – 1} + … + a_1 x^1 + a_0 \).
with real values for \(a_i\) and the exponents are natural numbers. The derivative of this can be found by using derivatives of linear functions, or in other words, constant multiples. This is because when we find the derivative of only constants (not variables), we end up multiplying them by their respective coefficients in order to keep the original equation true. Therefore, if we need to find the derivative \(\mathrm{d}f(x)\) at a point \(x_0\), we would use this formula:
\( \mathrm{d}f(x) = a_{n – 1} \mathrm{d}x + a_{n – 2}\mathrm{d}x^2 + … + a_1 \mathrm{d}x^1 + a_0 \).
Rational Functions
A rational function has the form:
\( f (x) = \frac {p(x)}{q(x)}\) where \(p, q\) are polynomials. The derivative of this can be found by using derivatives of quotient functions, or in other words, constant differences. Just like before, when we find the derivative of only constants (not variables), we end up multiplying them by their respective coefficients in order to keep the original equation true. Therefore, if we need to find \(\mathrm{d}f(x)\) at a point \(x_0\), we would use this formula:
\( \mathrm{d}f(x) = \frac {p(x)}{q(x)} \left(\frac {\mathrm{d}q(x)}{\mathrm{d} x}\right)+ q'(x)\).
Now that you know how to derive these two types of derivatives, I’ll show you one example each so you can get an idea for how differentiation works by applying the steps to a real-life situation.
Polynomial Example
Find \(\mathrm{d}f(x)\) for \(f(x) = x^3 – 6x\).
\(\mathrm{d}f(x) = 3x^{2} – 6\) by using formula for constant multiples or derivatives of linear functions.
Rational Example
Find \(\mathrm{d}f(x)\) for \( f (x) = \frac {4}{3}\left(\frac {2}{3}\right)(5 + x)\).
\(\mathrm{d}f(x) = -\frac {4}{9}\left(\frac {2}{3}\right)x – \frac {1}{3}\) by using formula for constant differences or derivatives of quotients.
Remember, differentiation can be tricky sometimes, so if you end up having trouble, feel free to post your questions to Math Homework Help!
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