Question

Differentiate the following functions
y = (〖6x〗^3+〖4x〗^2+3)(〖2x〗^2-〖4x〗^(-2)+5)

Answer 1

Answer on Question #51644 – Math – Differential Calculus | Equations

Question

Differentiate the following functions

$$y = (\llbracket 6 x \rrbracket \wedge 3 + \llbracket 4 x \rrbracket \wedge 2 + 3) (\llbracket 2 x \rrbracket \wedge 2 - \llbracket 4 x \rrbracket \wedge (-2) + 5)$$

Solution

Method 1

$$y = \left( \left( (6x)^3 + (4x)^2 + 3 \right) \left( (2x)^2 - (4x)^{-2} + 5 \right) \right);$$

Let $u = (6x)^3 + (4x)^2 + 3$, and $v = (2x)^2 - (4x)^{-2} + 5$, then

$$\begin{array}{l} u' = \left( (6x)^3 + (4x)^2 + 3 \right)' = \left( (6x)^3 \right)' + \left( (4x)^2 \right)' + (3)' = 3(6x)^2 \cdot 6 + 2(4x) \cdot 4 + 0 \\ = 18(6x)^2 + 8(4x) \\ v' = \left( (2x)^2 - (4x)^{-2} + 5 \right)' = \left( (2x)^2 \right)' + \left( -(4x)^{-2} \right)' + (5)' \\ = 2(2x) \cdot 2 + (-)(-2)(4x)^{-3} \cdot 4 + 0 = 4(2x) + 8(4x)^{-3} \\ \end{array}$$ $$\begin{array}{l} y' = (uv)' = u'v + uv' = \left[ 18(6x)^2 + 8(4x) \right] \cdot \left[ (2x)^2 - (4x)^{-2} + 5 \right] + \left[ (6x)^3 + (4x)^2 + 3 \right] \cdot \\ \left[ 4(2x) + 8(4x)^{-3} \right] = 4320x^4 + 256x^3 + \frac{3}{8x^3} + 3240x^2 + 184x - \frac{27}{2}; \\ \end{array}$$

Method 2

$$\begin{array}{l} y' = \left( \left[ (6x)^3 + (4x)^2 + 3 \right] \cdot \left[ (2x)^2 - (4x)^{-2} + 5 \right] \right)' \\ = \left( \left( (6x)^3(2x)^2 - (6x)^3(4x)^{-2} + 5(6x)^3 + (4x)^2(2x)^2 - (4x)^2(4x)^{-2} \right. \right. \\ \left. \left. + 5(4x)^2 + 3(2x)^2 - 3(4x)^{-2} + 15 \right) \right) = \\ \end{array}$$ $$\begin{array}{l} = \left( 6^3 2^2 x^5 - 6^3 4^{-2} x + 5 \cdot 6^3 x^3 + 4^2 2^2 x^4 - 1 + 5 \cdot 4^2 x^2 + 3 \cdot 2^2 x^2 - 3 \cdot 4^{-2} x^{-2} + 15 \right)' \\ = 5 \cdot 6^3 2^2 x^4 - 6^3 4^{-2} + 5 \cdot 6^3 \cdot 3x^2 + 4^2 2^2 \cdot 4x^3 + 5 \cdot 4^2 \cdot 2x + 3 \cdot 2^2 \cdot 2x - 3 \cdot 4^{-2} (-2) x^{-3} = \\ = 4320x^4 + 256x^3 + 3240x^2 + 184x - \frac{27}{2} + \frac{3}{8x^3} \\ \end{array}$$

Answer: $y' = 4320x^4 + 256x^3 + \frac{3}{8x^3} + 3240x^2 + 184x - \frac{27}{2}$

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