Solve 2x-3lceil-5/-33/2rceil

$$2x-3 \lceil \frac{ -5 }{ -3 } \frac{ 3 }{ 2 } \rceil$$

Evaluate

$$2x-9$$

Solution Steps

Fraction \frac{-5}{-3} can be simplified to \frac{5}{3} by removing the negative sign from both the numerator and the denominator.

$$2x-3\lceil \frac{5}{3}\times \frac{3}{2}\rceil$$

Multiply \frac{5}{3} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$2x-3\lceil \frac{5\times 3}{3\times 2}\rceil$$

Cancel out 3 in both numerator and denominator.

$$2x-3\lceil \frac{5}{2}\rceil$$

Dividing 5 by 2 gives 2 and remainder 1. Rewrite \frac{5}{2} as 2+\frac{1}{2}.

$$2x-3\lceil 2+\frac{1}{2}\rceil$$

The ceiling of a real number a is the smallest integer number greater than or equal to a. The ceiling of 2+\frac{1}{2} is 3.

$$2x-3\times 3$$

Multiply 3 and 3 to get 9.

$$2x-9$$

Differentiate w.r.t. x

$$2$$

Steps Using Derivative Rule for Sum

Fraction \frac{-5}{-3} can be simplified to \frac{5}{3} by removing the negative sign from both the numerator and the denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x-3\lceil \frac{5}{3}\times \frac{3}{2}\rceil )$$

Multiply \frac{5}{3} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x-3\lceil \frac{5\times 3}{3\times 2}\rceil )$$

Cancel out 3 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x-3\lceil \frac{5}{2}\rceil )$$

Dividing 5 by 2 gives 2 and remainder 1. Rewrite \frac{5}{2} as 2+\frac{1}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x-3\lceil 2+\frac{1}{2}\rceil )$$

The ceiling of a real number a is the smallest integer number greater than or equal to a. The ceiling of 2+\frac{1}{2} is 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x-3\times 3)$$

Multiply 3 and 3 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x-9)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$2x^{1-1}$$

Subtract 1 from 1.

$$2x^{0}$$

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

$$2\times 1$$

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

$$2$$