Solve (a^4)^-3diva^-1

$$( a ^ { 4 } ) ^ { - 3 } \div a ^ { - 1 }$$

Evaluate

$$\frac{1}{a^{11}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(a^{4}\right)^{-3}\times \frac{1}{\frac{1}{a}}$$

To raise a power to another power, multiply the exponents.

$$a^{4\left(-3\right)}a^{-\left(-1\right)}$$

Multiply 4 times -3.

$$a^{-12}a^{-\left(-1\right)}$$

Multiply -1 times -1.

$$a^{-12}a^{1}$$

To multiply powers of the same base, add their exponents.

$$a^{-12+1}$$

Add the exponents -12 and 1.

$$a^{-11}$$

Differentiate w.r.t. a

$$-\frac{11}{a^{12}}$$

Steps Using Chain Rule

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{-12}}{a^{-1}})$$

Rewrite a^{-1} as a^{-12}a^{11}. Cancel out a^{-12} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{11}})$$

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).

$$-\left(a^{11}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}a}(a^{11})$$

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}.

$$-\left(a^{11}\right)^{-2}\times 11a^{11-1}$$

Simplify.

$$-11a^{10}\left(a^{11}\right)^{-2}$$