Solve left|2left|xright|-5right|+x

$$\left| 2 \left| x \right| -5 \right| +x$$

Evaluate

$$11x$$

Solution Steps

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of 2 is 2.

$$2x|-5|+x$$

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of -5 is 5.

$$2x\times 5+x$$

Multiply 2 and 5 to get 10.

$$10x+x$$

Combine 10x and x to get 11x.

$$11x$$

Differentiate w.r.t. x

$$11$$

Steps Using Definition of a Derivative

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of 2 is 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x|-5|+x)$$

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of -5 is 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x\times 5+x)$$

Multiply 2 and 5 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}x}(10x+x)$$

Combine 10x and x to get 11x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(11x)$$

The derivative of ax^{n} is nax^{n-1}.

$$11x^{1-1}$$

Subtract 1 from 1.

$$11x^{0}$$

For any term t except 0, t^{0}=1.

$$11\times 1$$

For any term t, t\times 1=t and 1t=t.

$$11$$