5.3 Differentiating polynomialsIB Maths: Applications and Interpretation SL: Revision notes

The derivative of a function gives the gradient of its graph at every point. It is written $f'(x)$ or $\frac{dy}{dx}$ when $y=f(x)$. To find the gradient at a particular point, differentiate first and then substitute the $x$-value into the derivative. The derivative is also a rate of change: if $C(x)$ is a cost in AED and $x$ is the number of items, then $C'(x)$ is the rate of change of cost in AED per item.

If $f(x)=ax^n$, where $n$ is an integer, then $f'(x)=anx^{n-1}.$ Multiply by the power, then reduce the power by one. Special cases: $f(x)=kx$ gives $f'(x)=k$, and a constant $f(x)=k$ gives $f'(x)=0$ (a horizontal line has gradient zero). Examples: $\frac{d}{dx}(5x^4)=20x^3$; $\frac{d}{dx}(7x)=7$; $\frac{d}{dx}(9)=0$.

Differentiate a sum term by term: for $f(x)=ax^n+bx^{n-1}+\ldots$, differentiate each term separately and add the results. The power rule also works when $n$ is negative. First rewrite $\frac{1}{x^k}$ as $x^{-k}$. For $f(x)=\frac{3}{x^2}=3x^{-2}$, $f'(x)=-6x^{-3}=-\frac{6}{x^3}$. Subtracting one from a negative power makes it more negative. If the function is a product or a fraction, expand or divide first so that every term has the form $ax^n$. For $f(x)=(2x+1)(x-3)=2x^2-5x-3$, $f'(x)=4x-5$. For $f(x)=\frac{x^2+3}{x}=x+3x^{-1}$, $f'(x)=1-3x^{-2}$.

Worked example: $f(x)=2x^3-5x^2+x-4$. Then $f'(x)=6x^2-10x+1$.

• Gradient at $x=2$: $f'(2)=24-20+1=5$.
• Where is the gradient $1$? Solve $6x^2-10x+1=1$, so $2x(3x-5)=0$ and $x=0$ or $x=\frac{5}{3}$.
• Increasing or decreasing at a point: the sign of $f'$ at that point (positive means increasing). Always check you have found every solution: dividing both sides by $x$ would lose the solution $x=0$. Your GDC can solve $f'(x)=k$ directly.

Write the derivative on its own line with correct notation, for example $f'(x)=12x^2-12x+5$ or $\frac{dy}{dx}=\ldots$; do not write $f(x)=12x^2\ldots$. Final answers in context need units: a rate of change of cost in AED per item, or of surface area in cm$^2$ per cm. Give exact values or three significant figures. Check your work: each term's power should drop by one; a $x^1$ term becomes a constant; the constant term vanishes.