Solve a^-12/a^4*a^62

$$\frac { a ^ { - 12 } } { a ^ { 4 } \cdot a ^ { 62 } }$$

Evaluate

$$\frac{1}{a^{78}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 4 and 62 to get 66.

$$\frac{a^{-12}}{a^{66}}$$

Rewrite a^{66} as a^{-12}a^{78}. Cancel out a^{-12} in both numerator and denominator.

$$\frac{1}{a^{78}}$$

Differentiate w.r.t. a

$$-\frac{78}{a^{79}}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add 4 and 62 to get 66.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{-12}}{a^{66}})$$

Rewrite a^{66} as a^{-12}a^{78}. Cancel out a^{-12} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{78}})$$

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).

$$-\left(a^{78}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}a}(a^{78})$$

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}.

$$-\left(a^{78}\right)^{-2}\times 78a^{78-1}$$

Simplify.

$$-78a^{77}\left(a^{78}\right)^{-2}$$