Solve frac{x^{14}x^{5}(x^{3})^7}{x^18(x^7)^3}

$$\frac { x ^ { 14 } x ^ { 5 } ( x ^ { 3 } ) ^ { 7 } } { x ^ { 18 } ( x ^ { 7 } ) ^ { 3 } }$$

Evaluate

$$x$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 3 and 7 to get 21.

$$\frac{x^{14}x^{5}x^{21}}{x^{18}\left(x^{7}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 14 and 5 to get 19.

$$\frac{x^{19}x^{21}}{x^{18}\left(x^{7}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 19 and 21 to get 40.

$$\frac{x^{40}}{x^{18}\left(x^{7}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 3 to get 21.

$$\frac{x^{40}}{x^{18}x^{21}}$$

To multiply powers of the same base, add their exponents. Add 18 and 21 to get 39.

$$\frac{x^{40}}{x^{39}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 39 from 40 to get 1.

$$x^{1}$$

Calculate x to the power of 1 and get x.

$$x$$

Differentiate w.r.t. x

$$1$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 3 and 7 to get 21.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{14}x^{5}x^{21}}{x^{18}\left(x^{7}\right)^{3}})$$

To multiply powers of the same base, add their exponents. Add 14 and 5 to get 19.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{19}x^{21}}{x^{18}\left(x^{7}\right)^{3}})$$

To multiply powers of the same base, add their exponents. Add 19 and 21 to get 40.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{40}}{x^{18}\left(x^{7}\right)^{3}})$$

To raise a power to another power, multiply the exponents. Multiply 7 and 3 to get 21.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{40}}{x^{18}x^{21}})$$

To multiply powers of the same base, add their exponents. Add 18 and 21 to get 39.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{40}}{x^{39}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 39 from 40 to get 1.

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

Calculate x to the power of 1 and get x.

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

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

$$x^{1-1}$$

Subtract 1 from 1.

$$x^{0}$$

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

$$1$$