Solve (a^5)^2/(a^3)^4

$$\frac { ( a ^ { 5 } ) ^ { 2 } } { ( a ^ { 3 } ) ^ { 4 } }$$

Evaluate

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

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\frac{a^{10}}{\left(a^{3}\right)^{4}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 4 to get 12.

$$\frac{a^{10}}{a^{12}}$$

Rewrite a^{12} as a^{10}a^{2}. Cancel out a^{10} in both numerator and denominator.

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

Differentiate w.r.t. a

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

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{10}}{\left(a^{3}\right)^{4}})$$

To raise a power to another power, multiply the exponents. Multiply 3 and 4 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{10}}{a^{12}})$$

Rewrite a^{12} as a^{10}a^{2}. Cancel out a^{10} in both numerator and denominator.

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

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).

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

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}.

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

Simplify.

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

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

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