Derivative of $$$x^{3} + 5 x^{2} + 7 x + 4$$$
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Your Input
Find $$$\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right)$$$.
Solution
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}\left(\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{3}\right) + \frac{d}{dx} \left(5 x^{2}\right) + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(4\right)\right)}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 3$$$:
$${\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(5 x^{2}\right) = {\color{red}\left(3 x^{2}\right)} + \frac{d}{dx} \left(4\right) + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(5 x^{2}\right)$$The derivative of a constant is $$$0$$$:
$$3 x^{2} + {\color{red}\left(\frac{d}{dx} \left(4\right)\right)} + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(5 x^{2}\right) = 3 x^{2} + {\color{red}\left(0\right)} + \frac{d}{dx} \left(7 x\right) + \frac{d}{dx} \left(5 x^{2}\right)$$Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 5$$$ and $$$f{\left(x \right)} = x^{2}$$$:
$$3 x^{2} + {\color{red}\left(\frac{d}{dx} \left(5 x^{2}\right)\right)} + \frac{d}{dx} \left(7 x\right) = 3 x^{2} + {\color{red}\left(5 \frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(7 x\right)$$Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 7$$$ and $$$f{\left(x \right)} = x$$$:
$$3 x^{2} + {\color{red}\left(\frac{d}{dx} \left(7 x\right)\right)} + 5 \frac{d}{dx} \left(x^{2}\right) = 3 x^{2} + {\color{red}\left(7 \frac{d}{dx} \left(x\right)\right)} + 5 \frac{d}{dx} \left(x^{2}\right)$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$3 x^{2} + 7 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + 5 \frac{d}{dx} \left(x^{2}\right) = 3 x^{2} + 7 {\color{red}\left(1\right)} + 5 \frac{d}{dx} \left(x^{2}\right)$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:
$$3 x^{2} + 5 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + 7 = 3 x^{2} + 5 {\color{red}\left(2 x\right)} + 7$$Simplify:
$$3 x^{2} + 10 x + 7 = \left(x + 1\right) \left(3 x + 7\right)$$Thus, $$$\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right) = \left(x + 1\right) \left(3 x + 7\right)$$$.
Answer
$$$\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right) = \left(x + 1\right) \left(3 x + 7\right)$$$A