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

The calculator will find the derivative of x3+5x2+7x+4x^{3} + 5 x^{2} + 7 x + 4, with steps shown.

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Your Input

Find ddx(x3+5x2+7x+4)\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:

(ddx(x3+5x2+7x+4))=(ddx(x3)+ddx(5x2)+ddx(7x)+ddx(4)){\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 ddx(xn)=nxn1\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} with n=3n = 3:

(ddx(x3))+ddx(4)+ddx(7x)+ddx(5x2)=(3x2)+ddx(4)+ddx(7x)+ddx(5x2){\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 00:

3x2+(ddx(4))+ddx(7x)+ddx(5x2)=3x2+(0)+ddx(7x)+ddx(5x2)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 ddx(cf(x))=cddx(f(x))\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right) with c=5c = 5 and f(x)=x2f{\left(x \right)} = x^{2}:

3x2+(ddx(5x2))+ddx(7x)=3x2+(5ddx(x2))+ddx(7x)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 ddx(xn)=nxn1\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} with n=2n = 2:

3x2+5(ddx(x2))+ddx(7x)=3x2+5(2x)+ddx(7x)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 ddx(cf(x))=cddx(f(x))\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right) with c=7c = 7 and f(x)=xf{\left(x \right)} = x:

3x2+10x+(ddx(7x))=3x2+10x+(7ddx(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 ddx(xn)=nxn1\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} with n=1n = 1, in other words, ddx(x)=1\frac{d}{dx} \left(x\right) = 1:

3x2+10x+7(ddx(x))=3x2+10x+7(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:

3x2+10x+7=(x+1)(3x+7)3 x^{2} + 10 x + 7 = \left(x + 1\right) \left(3 x + 7\right)

Thus, ddx(x3+5x2+7x+4)=(x+1)(3x+7)\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right) = \left(x + 1\right) \left(3 x + 7\right).

Answer

ddx(x3+5x2+7x+4)=(x+1)(3x+7)\frac{d}{dx} \left(x^{3} + 5 x^{2} + 7 x + 4\right) = \left(x + 1\right) \left(3 x + 7\right)A