Solve (16-25x)-(50+{x}^frac{3{2}}

$$(16-25x)-(50+ { x }^{ \frac{ 3 }{ 2 } }$$

Evaluate

$$-x^{\frac{3}{2}}-25x-34$$

Solution Steps

To find the opposite of 50+x^{\frac{3}{2}}, find the opposite of each term.

$$16-25x-50-x^{\frac{3}{2}}$$

Subtract 50 from 16 to get -34.

$$-34-25x-x^{\frac{3}{2}}$$

Differentiate w.r.t. x

$$-\frac{3\sqrt{x}}{2}-25$$

Steps Using Derivative Rule for Sum

To find the opposite of 50+x^{\frac{3}{2}}, find the opposite of each term.

$$\frac{\mathrm{d}}{\mathrm{d}x}(16-25x-50-x^{\frac{3}{2}})$$

Subtract 50 from 16 to get -34.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-34-25x-x^{\frac{3}{2}})$$

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}.

$$-25x^{1-1}+\frac{3}{2}\left(-1\right)x^{\frac{3}{2}-1}$$

Subtract 1 from 1.

$$-25x^{0}+\frac{3}{2}\left(-1\right)x^{\frac{3}{2}-1}$$

Multiply \frac{3}{2} times -1.

$$-25x^{0}-\frac{3}{2}x^{\frac{3}{2}-1}$$

Subtract 1 from \frac{3}{2}.

$$-25x^{0}-\frac{3}{2}\sqrt{x}$$

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

$$-25-\frac{3}{2}\sqrt{x}$$