Derivative of x33x2x^{3} - 3 x^{2}

The calculator will find the derivative of x33x2x^{3} - 3 x^{2}, with steps shown.

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Your Input

Find ddx(x33x2)\frac{d}{dx} \left(x^{3} - 3 x^{2}\right).

Solution

The derivative of a sum/difference is the sum/difference of derivatives:

(ddx(x33x2))=(ddx(x3)ddx(3x2)){\color{red}\left(\frac{d}{dx} \left(x^{3} - 3 x^{2}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{3}\right) - \frac{d}{dx} \left(3 x^{2}\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(3x2)=(3x2)ddx(3x2){\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} - \frac{d}{dx} \left(3 x^{2}\right) = {\color{red}\left(3 x^{2}\right)} - \frac{d}{dx} \left(3 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=3c = 3 and f(x)=x2f{\left(x \right)} = x^{2}:

3x2(ddx(3x2))=3x2(3ddx(x2))3 x^{2} - {\color{red}\left(\frac{d}{dx} \left(3 x^{2}\right)\right)} = 3 x^{2} - {\color{red}\left(3 \frac{d}{dx} \left(x^{2}\right)\right)}

Apply the power rule ddx(xn)=nxn1\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} with n=2n = 2:

3x23(ddx(x2))=3x23(2x)3 x^{2} - 3 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = 3 x^{2} - 3 {\color{red}\left(2 x\right)}

Simplify:

3x26x=3x(x2)3 x^{2} - 6 x = 3 x \left(x - 2\right)

Thus, ddx(x33x2)=3x(x2)\frac{d}{dx} \left(x^{3} - 3 x^{2}\right) = 3 x \left(x - 2\right).

Answer

ddx(x33x2)=3x(x2)\frac{d}{dx} \left(x^{3} - 3 x^{2}\right) = 3 x \left(x - 2\right)A