Easy Way to Do Derivatives of Polynomials

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Differentiation is one of the fundamental processes in calculus. Differentiating a function (usually called f(x)) results in another function called the derivative, written as f'(x) ("f prime of x"). This derivative has many uses in physics and mathematics. For instance, if we graph a polynomial f(x), the derivative f'(x) tells us the slope (the rate of change) of the original function at all its points. The first section of this article teaches you to differentiate each term of the polynomial, one at a time. The second section uses this approach to walk through a typical example problem, differentiating an entire polynomial. After some practice, differentiating $5x^{3}$ will be as second nature as multiplying and dividing.

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Differentiate any constant to zero. A constant is any ordinary number, with no variable involved—for example, 3, -16, or $2/3$ . These are freebies in any differentiation problem, because their derivative is always 0. Just cross out that term and move on.^[1]
- Write this in the form ${\frac {\mathrm {d} }{\mathrm {d} x}}(3)=0$ . This says "The derivative of 3 with respect to x is 0."
- The derivative of a term is the "rate of change" of that term: how quickly that term changes inside a function. Since a constant never changes (3 will always stay 3), its rate of change is always zero.
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Multiply by the coefficient from the original term. The coefficient in front of the variable doesn't change when you differentiate the term. If you end up with more than one coefficient in your answer, multiply them together.^[7]

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Treat each term as a separate problem. Polynomials contain multiple terms, added or subtracted together. To differentiate the polynomial, differentiate each term separately. You can leave all of the addition and subtraction symbols alone.^[8]
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Get rid of the constant term. If there is a constant (a term without a variable), delete it. Differentiating always removes the constant term.^[9]
- In our example, 6 is the constant. ${\frac {\mathrm {d} }{\mathrm {d} x}}(6)=0$ , so we can get rid of it.
- Careful: only terms with no variables are constants. This rule does not affect numbers that are multiplied by x or any other variable.
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Move each variable's exponent to the front of the term. Remember, when we differentiate, each variable's exponent becomes a coefficient. If there is already a coefficient in front of the term, multiply the two coefficients together.^[10]
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Lower each exponent by one degree. To do this, subtract 1 from each exponent in each variable term.^[11]
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Find the value of the new equation at a given "x" value. You're already done with the differentiation, but there's a common next step in test problems. If you're asked to "evaluate the expression" for a value of x, all you need to do is replace each x in the new equation with the given value and solve.^[12]

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Question

How can I subtract and add polynomials?

Add or subtract "like" terms only, meaning you add or subtract their coefficients. Like terms are those having identical variables. For example, 5x²y³ and 10x²y³ are like terms, but 5x²y³ and 10x²y² are not.
Question

How can I go about finding the real roots of polynomial functions?

It depends on the degree of the polynomial. If it's linear, simply divide. If it's quadratic, use the formula x = (-b +/-√(b2 - 4ac))/2a. Cubic and quartic equations also have formulas to find the roots (although more complicated). However, it has been proven that there is no general, explicit formula for the roots of equations of degree 5 or higher (see Abel-Ruffini theorem). So you must try to factorize or use trial-and-error to find some roots. If they don't work, then by the above theorem you'll probably never find explicit solutions (unless you use elliptic functions). Of course, if you just want to analyze roots without getting actual values, just examine the graph with differentials.
Question

How do I evaluate 3^2 - 2^3 + 10x^0?

3² = 9. 2³ = 8. 10(x)^0 = 10(1) = 10. 9 - 8 + 10 = 11.
Question

If the exponents of variables is negative, then what?

A negative exponent indicates that the variable (accompanied by a positive rather than negative exponent) is being divided into (rather than multiplied by) the rest of the expression. In contrast, if a variable with a negative exponent appears in the denominator of an expression, that indicates that the variable (accompanied by a positive exponent) is being multiplied by the rest of the expression.
Question

How can I simplify radical expression (2 square root of x)(3 cube root of x)?

That expression can be written as (2)(x^½)(3)(x^1/3) = (6)(x^5/6) or 6 times the sixth root of x^5.