For $f$ meromorphic in the complex plane and $\varphi$ meromorphic in the unit disc, sharp upper bounds are obtained for $$m\left(r,\frac{f^{(k)}}{f^{(j)}}\right)=\frac{1}{2\pi}\int_0^{2\pi} \log^{+}\left|\frac{f^{(k)}(re^{i\theta})}{f^{(j)}(re^{i\theta})}\right |\, d\theta,\qquad r<\infty,$$ and $$m\left(r,\frac{\vp^{(k)}}{\vp^{(j)}}\right)=\frac{1}{2\pi}\int_0^{2\pi} \log^{+}\left|\frac{\vp^{(k)}(re^{i\theta})}{\vp^{(j)}(re^{i\theta})}\right |\, d\theta,\qquad r<1,$$ where $k$ and $j$ are integers satisfying $k>j\geq 0$. The results generalize the logarithmic derivative estimate due to Gol'dberg and Grinshtein to derivatives of higher order.