Solve frac{(a^{3})^{2}*a^-2diva^4}{a^2*a^3}=

$$\frac { ( a ^ { 3 } ) ^ { 2 } \times a ^ { - 2 } \div a ^ { 4 } } { a ^ { 2 } \times a ^ { 3 } } =$$

Evaluate

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

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\frac{a^{6}a^{-2}}{a^{4}}}{a^{2}a^{3}}$$

To multiply powers of the same base, add their exponents. Add 6 and -2 to get 4.

$$\frac{\frac{a^{4}}{a^{4}}}{a^{2}a^{3}}$$

Divide a^{4} by a^{4} to get 1.

$$\frac{1}{a^{2}a^{3}}$$

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

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

Differentiate w.r.t. a

$$-\frac{5}{a^{6}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{6}a^{-2}}{a^{4}}}{a^{2}a^{3}})$$

To multiply powers of the same base, add their exponents. Add 6 and -2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{4}}{a^{4}}}{a^{2}a^{3}})$$

Divide a^{4} by a^{4} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{2}a^{3}})$$

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

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

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(a^{5}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}a}(a^{5})$$

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}.

$$-\left(a^{5}\right)^{-2}\times 5a^{5-1}$$

Simplify.

$$-5a^{4}\left(a^{5}\right)^{-2}$$