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(d)/(dw)((x^(2)+3x)^(4)+2x)^(2)

(d)/(dw)\left(\left(x^{ $2$ }+ $3$ x\right)^{ $4$ }+ $2$ x\right)^{ $2$ }

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Find derivatives of using multiple formulae

Full solution

Q. (d)/(dw)\left(\left(x^{

2

}+

3

x\right)^{

4

}+

2

x\right)^{

2

}

Distribute the derivative: Distribute the derivative. $\newline$ $\frac{d}{dw} \left( (x^2 + 3x)^4 + (2x)^2 \right) = \frac{d}{dw} \left( (x^2 + 3x)^4 \right) + \frac{d}{dw} \left( (2x)^2 \right)$
Derivative of (x^ $2$ + $3$ x)^ $4$ : Derivative of $(x^2 + 3x)^4$ . $\newline$ $\frac{d}{dw} \left( (x^2 + 3x)^4 \right) = 4(x^2 + 3x)^3 \cdot \frac{d}{dw} (x^2 + 3x)$
Derivative of x^ $2$ + $3$ x: Derivative of $x^2 + 3x$ with respect to $w$ . $\newline$ $\frac{d}{dw} (x^2 + 3x) = 2x + 3$
Combine results from Step $2$ and Step $3$ : Combine the results from Step $2$ and Step $3$ . $\newline$ $4(x^2 + 3x)^3 \cdot (2x + 3)$
Derivative of ( $2$ x)^ $2$ : Derivative of $(2x)^2$ . $\newline$ $\frac{d}{dw} \left( (2x)^2 \right) = \frac{d}{dw} (4x^2) = 8x$
Combine all results: Combine all the results. $\newline$ $\frac{d}{dw} \left( (x^2 + 3x)^4 + (2x)^2 \right) = 4(x^2 + 3x)^3 \cdot (2x + 3) + 8x$

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