Differentiation of Polynomials

\( \scriptsize If \; y = ax^2 + bx + c \), 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 \( \; – \frac{3}{2x^{\scriptsize \frac{3}{2}}} \)