Solve a^2a/(a^3)^2=

$$\frac { a ^ { 2 } a } { ( a ^ { 3 } ) ^ { 2 } } =$$

Evaluate

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

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

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

$$\frac{a^{3}}{a^{6}}$$

Rewrite a^{6} as a^{3}a^{3}. Cancel out a^{3} in both numerator and denominator.

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

Differentiate w.r.t. a

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

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

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

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

Rewrite a^{6} as a^{3}a^{3}. Cancel out a^{3} in both numerator and denominator.

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

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).

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

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}.

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

Simplify.

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