<prompt> Fumana equation bakeng sa motsu oa sebe x.nan  <response>: Instruction in English: Derive the equation for the derivative of sin x. . Response in English: The derivative of a function gives the slope of the tangent line to the function at any point. We can use the definition of derivative to derive the equation for the derivative of sinx.The definition of derivative of a function fx at a point x is given as:fx limh-0 fxh - fxhSo, let fx sinx. Then, fx limh-0 sinxh - sinxhUsing the trigonometric addition identity sinxy sinxcosy cosxsiny, we can expand sinxh as follows:fx limh-0 sinxcosh cosxsinh - sinxhSimplifying,fx limh-0 sinxcosh - 1 cosxsinhhSince h is approaching 0, we can use the small angle approximation for sinh and cosh, which state that for very small values of h, sinh is approximately equal to h, and cosh is approximately equal to 1. Hence, we can write:fx limh-0 sinx1 - 1 cosxhhfx limh-0 cosxThis simply equals to cosx. Therefore, the derivative of sinx is cosx.So the equation for the derivative of sin x is:dsin xdx cos x . Response in Sesotho: Motsoako oa tšebetso o fana ka moepa oa tangent ho ea tšebetsong ntlheng efe kapa efe. Re ka sebelisa tlhaloso ea morifi ho fumana equation bakeng sa derivative ea sebex. Tlhaloso ea motsoako oa lentsoe fx ntlheng ea x e fanoe e le: fx limh-0 fxh - fxh Kahoo, tlohela fx sebex. Ebe, fx limh-0 sinxh - sinxh Ho sebedisa trigonometric identity identity sinxy sinxcosy cosxsiny, re ka atolosa sebexh ka tsela e latelang: fx limh-0 sinxcosh cosx sebeh - sebexh Ho nolofatsa, fx limh-0 sebexcosh - 1 cosxsinh h Kaha h e atamela 0, re ka sebelisa tekanyo e nyenyane ea angle ea sinh le cosh, e bolelang hore bakeng sa litekanyetso tse nyenyane haholo tsa h, sinh e batla e lekana le h, le cos h e batla e lekana le 1. Kahoo, re ka ngola: fx limh-0 sinx1 - 1 cosxhh fx limh-0 cosx Sena se lekana feela le cosx. Ka hona, lentsoe le tsoang ho sebex ke cosx. Kahoo equation bakeng sa motsu oa sebe x ke: dsin xdx cos x 