Solve x/x^2{frac{frac{x^3/x^4/x^5}}{{x}^6}}

$$\frac{ x }{ \frac{ \frac{ { x }^{ 2 } }{ \frac{ \frac{ { x }^{ 3 } }{ { x }^{ 4 } } }{ { x }^{ 5 } } } }{ { x }^{ 6 } } }$$

Evaluate

$$\frac{1}{x}$$

Solution Steps

Rewrite x^{4} as x^{3}x. Cancel out x^{3} in both numerator and denominator.

$$\frac{x}{\frac{\frac{x^{2}}{\frac{\frac{1}{x}}{x^{5}}}}{x^{6}}}$$

Divide x by \frac{\frac{x^{2}}{\frac{\frac{1}{x}}{x^{5}}}}{x^{6}} by multiplying x by the reciprocal of \frac{\frac{x^{2}}{\frac{\frac{1}{x}}{x^{5}}}}{x^{6}}.

$$\frac{xx^{6}}{\frac{x^{2}}{\frac{\frac{1}{x}}{x^{5}}}}$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\frac{x^{7}}{\frac{x^{2}}{\frac{\frac{1}{x}}{x^{5}}}}$$

Divide x^{2} by \frac{\frac{1}{x}}{x^{5}} by multiplying x^{2} by the reciprocal of \frac{\frac{1}{x}}{x^{5}}.

$$\frac{x^{7}}{\frac{x^{2}x^{5}}{\frac{1}{x}}}$$

To multiply powers of the same base, add their exponents. Add 2 and 5 to get 7.

$$\frac{x^{7}}{\frac{x^{7}}{\frac{1}{x}}}$$

Divide x^{7} by \frac{1}{x} by multiplying x^{7} by the reciprocal of \frac{1}{x}.

$$\frac{x^{7}}{x^{7}x}$$

To multiply powers of the same base, add their exponents. Add 7 and 1 to get 8.

$$\frac{x^{7}}{x^{8}}$$

Cancel out x^{7} in both numerator and denominator.

$$\frac{1}{x}$$

Differentiate w.r.t. x

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

Steps Using Definition of a Derivative

Rewrite x^{4} as x^{3}x. Cancel out x^{3} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x}{\frac{\frac{x^{2}}{\frac{\frac{1}{x}}{x^{5}}}}{x^{6}}})$$

Divide x by \frac{\frac{x^{2}}{\frac{\frac{1}{x}}{x^{5}}}}{x^{6}} by multiplying x by the reciprocal of \frac{\frac{x^{2}}{\frac{\frac{1}{x}}{x^{5}}}}{x^{6}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{xx^{6}}{\frac{x^{2}}{\frac{\frac{1}{x}}{x^{5}}}})$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{7}}{\frac{x^{2}}{\frac{\frac{1}{x}}{x^{5}}}})$$

Divide x^{2} by \frac{\frac{1}{x}}{x^{5}} by multiplying x^{2} by the reciprocal of \frac{\frac{1}{x}}{x^{5}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{7}}{\frac{x^{2}x^{5}}{\frac{1}{x}}})$$

To multiply powers of the same base, add their exponents. Add 2 and 5 to get 7.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{7}}{\frac{x^{7}}{\frac{1}{x}}})$$

Divide x^{7} by \frac{1}{x} by multiplying x^{7} by the reciprocal of \frac{1}{x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{7}}{x^{7}x})$$

To multiply powers of the same base, add their exponents. Add 7 and 1 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{7}}{x^{8}})$$

Cancel out x^{7} in both numerator and denominator.

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

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

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

Subtract 1 from -1.

$$-x^{-2}$$