Derivative of x^3 + 5*x^2 + 7*x + 4

The calculator will find the derivative of

x^{3} + 5 x^{2} + 7 x + 4

, with steps shown.

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 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)} + \frac{d}{dx} \left(7 x\right) = 3 x^{2} + 5 {\color{red}\left(2 x\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} + 10 x + {\color{red}\left(\frac{d}{dx} \left(7 x\right)\right)} = 3 x^{2} + 10 x + {\color{red}\left(7 \frac{d}{dx} \left(x\right)\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} + 10 x + 7 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 3 x^{2} + 10 x + 7 {\color{red}\left(1\right)}

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

Derivative of x3+5x2+7x+4x^{3} + 5 x^{2} + 7 x + 4x3+5x2+7x+4

Your Input

Solution

Answer

Derivative of $x^{3} + 5 x^{2} + 7 x + 4$