Solve (81a-8)^-frac{3{4}}

$$( 81 a - 8 ) ^ { - \frac { 3 } { 4 } }$$

Differentiate w.r.t. a

$$-\frac{243}{4\left(81a-8\right)^{\frac{7}{4}}}$$

Steps Using Chain Rule

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).

$$-\frac{3}{4}\left(81a^{1}-8\right)^{-\frac{3}{4}-1}\frac{\mathrm{d}}{\mathrm{d}a}(81a^{1}-8)$$

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}.

$$-\frac{3}{4}\left(81a^{1}-8\right)^{-\frac{7}{4}}\times 81a^{1-1}$$

Simplify.

$$-\frac{243}{4}a^{0}\left(81a^{1}-8\right)^{-\frac{7}{4}}$$

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

$$-\frac{243}{4}a^{0}\left(81a-8\right)^{-\frac{7}{4}}$$

For any term t except 0, t^{0}=1.

$$-\frac{243}{4}\left(81a-8\right)^{-\frac{7}{4}}$$

Evaluate

$$\frac{1}{\left(81a-8\right)^{\frac{3}{4}}}$$