Derivative of x32xx^{3} - 2 x at x=cx = c

The calculator will find the derivative of x32xx^{3} - 2 x at x=cx = c, with steps shown.

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Your Input

Find ddx(x32x)\frac{d}{dx} \left(x^{3} - 2 x\right) and evaluate it at x=cx = c.

Solution

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

(ddx(x32x))=(ddx(x3)ddx(2x)){\color{red}\left(\frac{d}{dx} \left(x^{3} - 2 x\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{3}\right) - \frac{d}{dx} \left(2 x\right)\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=2c = 2 and f(x)=xf{\left(x \right)} = x:

(ddx(2x))+ddx(x3)=(2ddx(x))+ddx(x3)- {\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)} + \frac{d}{dx} \left(x^{3}\right) = - {\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(x^{3}\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))2ddx(x)=(3x2)2ddx(x){\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} - 2 \frac{d}{dx} \left(x\right) = {\color{red}\left(3 x^{2}\right)} - 2 \frac{d}{dx} \left(x\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:

3x22(ddx(x))=3x22(1)3 x^{2} - 2 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 3 x^{2} - 2 {\color{red}\left(1\right)}

Thus, ddx(x32x)=3x22\frac{d}{dx} \left(x^{3} - 2 x\right) = 3 x^{2} - 2.

Finally, evaluate the derivative at x=cx = c.

(ddx(x32x))(x=c)=3c22\left(\frac{d}{dx} \left(x^{3} - 2 x\right)\right)|_{\left(x = c\right)} = 3 c^{2} - 2

Answer

ddx(x32x)=3x22\frac{d}{dx} \left(x^{3} - 2 x\right) = 3 x^{2} - 2A

(ddx(x32x))(x=c)=3c22\left(\frac{d}{dx} \left(x^{3} - 2 x\right)\right)|_{\left(x = c\right)} = 3 c^{2} - 2A