Introduction to Derivatives

How do you differentiate a polynomial?

Quick answer

Differentiating a polynomial involves applying the power rule to each term individually. Multiply each coefficient by its variable's current exponent, reduce that exponent by one, and change any constant terms to zero. Because sums differentiate term by term, a polynomial of degree n always differentiates into a polynomial of degree n minus one.

The topic explained

Differentiate term by term with the power rule

When differentiating polynomials, take each term one at a time. The power rule states that the derivative of x^n is n * x^(n - 1). You bring the existing exponent down to multiply the term, then reduce the power by one. When a term has a constant coefficient in front, keep it in place and multiply it by the incoming exponent.

Constant terms on their own differentiate to zero because a constant value never changes, making its instantaneous rate of change zero. Linear terms such as -7x simplify to their constant coefficient, -7, because x^1 becomes 1 * x^0, and x^0 equals 1.

Step-by-step example

Consider the function f(x) = x^3 + 4x^2 - 7x + 2. Differentiate each piece separately.

First, x^3 becomes 3x^2. Second, 4x^2 becomes 4 * 2x^1, which simplifies to 8x. Third, -7x becomes -7. Finally, the constant 2 becomes 0. Combining these results yields the derivative f'(x) = 3x^2 + 8x - 7.

Notice the degree of the expression: the original polynomial was a cubic (degree 3), and its derivative is a quadratic (degree 2). Checking that the highest power dropped by exactly one is a quick way to catch missing terms.

Evaluating the slope at a specific point

The derivative function gives the slope of the tangent line at any point along the curve. To find the slope at a particular x-value, differentiate the polynomial first, then plug that value into your resulting derivative expression.

For example, to find the slope of f(x) = x^2 - 6x at x = 4, first find the derivative: f'(x) = 2x - 6. Then substitute 4 for x: 2(4) - 6 = 8 - 6 = 2. The slope of the tangent line to the curve at x = 4 is 2.

Mistakes to watch for

The most frequent slip is carrying a standalone constant through to the derivative instead of setting it to zero. If you start with a quadratic and end with a quadratic, look for an undifferentiated constant.

Another error is lowering the exponent without multiplying by the original power, turning x^3 into x^2 instead of 3x^2. Finally, remember that while sums and differences differentiate term by term, multiplied factors do not; expressions multiplied together require the product rule.

Free printable worksheet

Differentiating Polynomial Functions Practice

Differentiate each polynomial with respect to x. For the final questions, evaluate the derivative at the given coordinate.

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Differentiating Polynomial Functions PracticeName:

Find f'(x) for f(x) = x^5
Find f'(x) for f(x) = 6x^3
Find f'(x) for f(x) = 3x^2 - 5x + 9
Find f'(x) for f(x) = -2x^4 + 7x^3 - x + 14
Find f'(x) for f(x) = (1/3)x^3 - (1/2)x^2 + 4x - 11
Find the slope of the tangent to f(x) = x^2 - 6x at x = 4
Find the slope of the tangent to f(x) = 2x^3 - 3x^2 + 1 at x = -1

Common questions

Why does a constant term differentiate to zero?

A derivative measures the rate at which an output changes relative to an input. A constant never changes, so its rate of change is zero. On a graph, y = c is a flat horizontal line with a slope of zero everywhere.

Can I use this method if terms are multiplied together?

No. Sums and differences can be differentiated term by term, but products cannot. If two polynomial factors are multiplied, either expand the brackets first to create a standard polynomial or use the product rule.

How do you know if you made an error in the degree?

Differentiating a polynomial always decreases the degree of the highest term by exactly one. If your original function starts as a 4th-degree polynomial, your derivative must be a 3rd-degree polynomial. If the degree did not decrease, a term was left undifferentiated.

Does the power rule work for negative powers or roots?

Yes. The power rule works for any real exponent. Roots can be rewritten as fractional exponents such as x^(1/2), and reciprocals can be rewritten with negative exponents such as x^(-1), before applying the rule.

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