Solve a^{-frac{1/9}*[a^-frac{1{3}}]^2}{-a^2}div[-1/a]^2

$$\frac { a ^ { - \frac { 1 } { 9 } } \cdot [ a ^ { - \frac { 1 } { 3 } } ] ^ { 2 } } { - a ^ { 2 } } \div [ - \frac { 1 } { a } ] ^ { 2 }$$

Evaluate

$$-\frac{1}{a^{\frac{7}{9}}}$$

Short Solution Steps

To raise a power to another power, multiply the exponents. Multiply -\frac{1}{3} and 2 to get -\frac{2}{3}.

$$\frac{\frac{a^{-\frac{1}{9}}a^{-\frac{2}{3}}}{-a^{2}}}{\left(-\frac{1}{a}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add -\frac{1}{9} and -\frac{2}{3} to get -\frac{7}{9}.

$$\frac{\frac{a^{-\frac{7}{9}}}{-a^{2}}}{\left(-\frac{1}{a}\right)^{2}}$$

Calculate -\frac{1}{a} to the power of 2 and get \left(\frac{1}{a}\right)^{2}.

$$\frac{\frac{a^{-\frac{7}{9}}}{-a^{2}}}{\left(\frac{1}{a}\right)^{2}}$$

To raise \frac{1}{a} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{a^{-\frac{7}{9}}}{-a^{2}}}{\frac{1^{2}}{a^{2}}}$$

Divide \frac{a^{-\frac{7}{9}}}{-a^{2}} by \frac{1^{2}}{a^{2}} by multiplying \frac{a^{-\frac{7}{9}}}{-a^{2}} by the reciprocal of \frac{1^{2}}{a^{2}}.

$$\frac{a^{-\frac{7}{9}}a^{2}}{\left(-a^{2}\right)\times 1^{2}}$$

Factor the expressions that are not already factored.

$$\frac{a^{-\frac{7}{9}}a^{2}}{-1^{2}a^{2}}$$

Cancel out a^{2} in both numerator and denominator.

$$\frac{a^{-\frac{7}{9}}}{-1^{2}}$$

Expand the expression.

$$-a^{-\frac{7}{9}}$$

Differentiate w.r.t. a

$$\frac{7}{9a^{\frac{16}{9}}}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply -\frac{1}{3} and 2 to get -\frac{2}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{-\frac{1}{9}}a^{-\frac{2}{3}}}{-a^{2}}}{\left(-\frac{1}{a}\right)^{2}})$$

To multiply powers of the same base, add their exponents. Add -\frac{1}{9} and -\frac{2}{3} to get -\frac{7}{9}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{-\frac{7}{9}}}{-a^{2}}}{\left(-\frac{1}{a}\right)^{2}})$$

Calculate -\frac{1}{a} to the power of 2 and get \left(\frac{1}{a}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{-\frac{7}{9}}}{-a^{2}}}{\left(\frac{1}{a}\right)^{2}})$$

To raise \frac{1}{a} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{-\frac{7}{9}}}{-a^{2}}}{\frac{1^{2}}{a^{2}}})$$

Divide \frac{a^{-\frac{7}{9}}}{-a^{2}} by \frac{1^{2}}{a^{2}} by multiplying \frac{a^{-\frac{7}{9}}}{-a^{2}} by the reciprocal of \frac{1^{2}}{a^{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{-\frac{7}{9}}a^{2}}{\left(-a^{2}\right)\times 1^{2}})$$

Factor the expressions that are not already factored in \frac{a^{-\frac{7}{9}}a^{2}}{\left(-a^{2}\right)\times 1^{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{-\frac{7}{9}}a^{2}}{-1^{2}a^{2}})$$

Cancel out a^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{-\frac{7}{9}}}{-1^{2}})$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{-\frac{7}{9}}}{-1})$$

Anything divided by -1 gives its opposite.

$$\frac{\mathrm{d}}{\mathrm{d}a}(-a^{-\frac{7}{9}})$$

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

$$-\frac{7}{9}\left(-1\right)a^{-\frac{7}{9}-1}$$

Multiply -\frac{7}{9} times -1.

$$\frac{7}{9}a^{-\frac{7}{9}-1}$$

Subtract 1 from -\frac{7}{9}.

$$\frac{7}{9}a^{-\frac{16}{9}}$$