türevlerle

Türevlerle is a Turkish phrase meaning "with derivatives" or "by means of derivatives." In mathematical writing, türevlerle describes procedures, analyses, or arguments that rely on the derivatives of a function. The word combines türev (derivative) with the plural instrumental suffix -lerle, signaling the use of multiple derivatives or derivative-based methods.

In calculus and analysis, derivatives measure the instantaneous rate of change and provide local linear approximations of functions. Türevlerle approaches use these tools to study behavior, determine slopes, identify extrema, approximate functions via Taylor series, and implement numerical methods. Common contexts include optimization, where derivatives inform gradient and Hessian-based steps, and curve sketching, where sign changes and zeroes of derivatives indicate turning points and inflection points. In numerical analysis, algorithms such as Newton’s method or quasi-Newton methods rely on derivative information, i.e., they operate türevlerle.

In higher dimensions, the phrase can be extended to reference gradients, Jacobians, and Hessians as the differential information used in analysis. When referring specifically to partial derivatives, Turkish writers may use phrases like kısmi türevlerle to specify the type of derivatives involved.

Limitations of türevlerle methods include the need for differentiability in the region of interest and well-behaved derivatives. If a function is not differentiable or derivatives are difficult to obtain, derivative-free approaches may be preferred.

See also: derivative, differentiation, gradient, Hessian, Taylor series.