Differentiate polynomials

What changes when this term is differentiated?

Move the original power through all five values. Compare the old and new coefficient and power

When the original power is one, why does the derivative have no variable factor?

Two operations per term

The derivative of a polynomial is another polynomial. Differentiate each term and combine like powers. A constant becomes zero

1. 1Differentiate each power For a positive integer power, multiply the coefficient by that power and reduce the power by one \(\displaystyle \frac{d}{dx}(ax^n)=nax^{n-1},\quad n\ge1\)
2. 2A constant has zero derivative A constant does not change as the input changes. Keep a zero entry when the response asks for the constant coefficient \(\displaystyle \frac{d}{dx}(c)=0\)
3. 3Keep addition and scales Differentiate a sum term by term. A fixed rational scale multiplies the derivative too \(\displaystyle \frac{d}{dx}(af(x)+bg(x))=af\prime(x)+bg\prime(x)\)

Given

\(\displaystyle f(x)=-\frac23x^3+5\)

Enter the coefficient of this power in the derivative

Given

\(\displaystyle f(x)=7\)

Enter the constant coefficient of the derivative

Given

\(\displaystyle f(x)=2x^4-3x+5\)

Enter the coefficient of x in the derivative, including zero if absent

Before differentiating this product using the polynomial rule, which step is valid?

Expand or cancel first

These questions require only polynomial differentiation. Carry out the algebra before applying the rule. Cancellation preserves the original nonzero-input restriction

1. 1Do the algebra first Expand products and scaled brackets, or cancel common factors under the original input restriction. Then apply the polynomial power rule
2. 2Keep the original domain Cancelling an input factor does not define the original function at an excluded input

Given

\(\displaystyle f(x)=\frac{2x^3-5x}{x},\quad x\ne0\)

After simplifying the displayed rule, enter the coefficient of x in its derivative

Which derivative notation matches this rule, whose input is t?

Given

\(\displaystyle y=\frac12(4t^3-6t+8)\)

Enter the constant coefficient of the derivative