Solve x^3*x^0*x^-5divx^2

$$x ^ { 3 } \cdot x ^ { 0 } \cdot x ^ { - 5 } \div x ^ { 2 }$$

Evaluate

$$\frac{1}{x^{4}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 0 to get 3.

$$\frac{x^{3}x^{-5}}{x^{2}}$$

To multiply powers of the same base, add their exponents. Add 3 and -5 to get -2.

$$\frac{x^{-2}}{x^{2}}$$

Rewrite x^{2} as x^{-2}x^{4}. Cancel out x^{-2} in both numerator and denominator.

$$\frac{1}{x^{4}}$$

Differentiate w.r.t. x

$$-\frac{4}{x^{5}}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{0}}{x^{5}}x^{3-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{5}}x^{1})$$

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}.

$$\frac{1}{x^{5}}x^{1-1}$$

Do the arithmetic.

$$\frac{1}{x^{5}}x^{0}$$

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

$$\frac{1}{x^{5}}\times 1$$

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

$$\frac{1}{x^{5}}$$