Solve (m+7)^-frac{1{6}}(m+7)^-frac{2{3}}

$$( m + 7 ) ^ { - \frac { 1 } { 6 } } ( m + 7 ) ^ { - \frac { 2 } { 3 } }$$

Differentiate w.r.t. m

$$-\frac{5}{6\left(m+7\right)^{\frac{11}{6}}}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add -\frac{1}{6} and -\frac{2}{3} to get -\frac{5}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\left(m+7\right)^{-\frac{5}{6}})$$

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).

$$-\frac{5}{6}\left(m^{1}+7\right)^{-\frac{5}{6}-1}\frac{\mathrm{d}}{\mathrm{d}m}(m^{1}+7)$$

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}.

$$-\frac{5}{6}\left(m^{1}+7\right)^{-\frac{11}{6}}m^{1-1}$$

Simplify.

$$-\frac{5}{6}m^{0}\left(m^{1}+7\right)^{-\frac{11}{6}}$$

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

$$-\frac{5}{6}m^{0}\left(m+7\right)^{-\frac{11}{6}}$$

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

$$-\frac{5}{6}\left(m+7\right)^{-\frac{11}{6}}$$

Evaluate

$$\frac{1}{\left(m+7\right)^{\frac{5}{6}}}$$