2.2 First Derivative for Polynomial Function

2.2 First Derivative for Polynomial Function

(A) Differentiating a Constant

(B) Differentiating Variable with Index n

(C) Differentiating a Linear Function

(D) Differentiating a Polynomial Function

(E) Differentiating Fractional Function

(F) Differentiating Square Root Function

Example:

Find

\frac{d y}{d x}

for each of the following functions:

(a) y= 12

(b) y= x⁴

(c) y= 3x

(d) y= 5x³

\begin{array}{l} (e) y = \frac{1}{x} \\ (f) y = \frac{2}{x^{4}} \\ (g) y = \frac{2}{5 x^{2}} \\ (h) y = 3 \sqrt{x} \\ (i) y = 4 \sqrt{x^{3}} \end{array}

Solution:

(a)
y = 12

\frac{d y}{d x} = 0

(b)
y = x⁴

\frac{d y}{d x}

= 4x³

(c)
y = 3x

\frac{d y}{d x}

= 3

(d)
y = 5x³

\frac{d y}{d x}

= 3(5x²) = 15x²

(e)

\begin{array}{l} y = \frac{1}{x} = x^{- 1} \\ \frac{d y}{d x} = - x^{- 1 - 1} = - \frac{1}{x^{2}} \end{array}

(f)

\begin{array}{l} y = \frac{2}{x^{4}} = 2 x^{- 4} \\ \frac{d y}{d x} = - 4 (2 x^{- 4 - 1}) = - 8 x^{- 5} = - \frac{8}{x^{5}} \end{array}

(g)

\begin{array}{l} y = \frac{2}{5 x^{2}} = \frac{2 x^{- 2}}{5} \\ \frac{d y}{d x} = - 2 (\frac{2 x^{- 2 - 1}}{5}) = - \frac{4 x^{- 3}}{5} = - \frac{4}{5 x^{3}} \end{array}

(h)

\begin{array}{l} y = 3 \sqrt{x} = 3 {(x)}^{\frac{1}{2}} \\ \frac{d y}{d x} = \frac{1}{2} (3 x^{\frac{1}{2} - 1}) = \frac{3}{2} x^{- \frac{1}{2}} = \frac{3}{2 \sqrt{x}} \end{array}

(i)

\begin{array}{l} y = 4 \sqrt{x^{3}} = 4 {(x^{3})}^{\frac{1}{2}} = 4 x^{\frac{3}{2}} \\ \frac{d y}{d x} = \frac{3}{2} (4 x^{\frac{3}{2} - 1}) = 6 x^{\frac{1}{2}} = 6 \sqrt{x} \end{array}