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Lesson 5, Topic 4
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# Differentiation of Polynomials

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$$\scriptsize If \; y = ax^2 + bx + c$$, then $$\frac{dy}{dx} \scriptsize = 2a + b$$

The derivatives of the separate terms ax2, bx, c are 2ax, b and 0 respectively.

Hence the derivative of y is the sum of the separate derivatives. This is true for polynomial or collection of terms involving the powers of variables.

Example:

1. Differentiate with respect to x, y = 4x3 – x2 + 2x – 1

Solution:

$$\scriptsize y = 4x^{3} – x^{2} + 2x – 1$$

$$\frac{dy}{dx} \scriptsize = 3 \times 4x^{3 -1}\; – 2 \times x^{2 -1} + 1 \times 2x^{1 -1}\; – 0$$

$$\frac{dy}{dx} \scriptsize = 12x^{2} \; – 2x + 2$$

Example:

2. Find $$\frac{dy}{dx}\scriptsize \; if \; y = \normalsize \frac {3}{\sqrt{x}}$$

Express the function in index form, $$\scriptsize y = 3x^{-\frac{1}{2}}$$

$$\frac{dy}{dx} \scriptsize = 3 \times \left ( – \frac{1}{2} \right)x^{-\frac{1}{2} – 1}$$

$$\frac{dy}{dx} = \frac{-3}{2} \scriptsize x^{-\frac{3}{2}}$$

Which could also be written as $$\; – \frac{3}{2x^{\scriptsize \frac{3}{2}}}$$

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