微分三角函数
如何求导sin cos和tan?
- 的导数
" class="Wirisformula" role="math" style="vertical-align:-4px;height:20px;width:66px">是y = sin x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" style="vertical-align:-14px;height:41px;width:86px">d y d x = cos x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" style="vertical-align:-4px;height:19px;width:70px">是y = cos x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" style="vertical-align:-14px;height:41px;width:97px">d y d x = - sin x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" alt="开始mathsize 16px样式加粗空格加粗斜体y加粗等于加粗tan加粗空格加粗斜体x结束样式" style="vertical-align:-4px;height:19px;width:69px">是y = tan x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开始mathsize 16px样式加粗空格分数分子加粗d加粗斜体y除以分母加粗d加粗斜体x结束分数加粗等于分数分子加粗1除以分母加粗cos的加粗2加粗空格加粗x结束分数结束样式" style="vertical-align:-16px;height:42px;width:102px">d y d x = 1 cos 2 x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 这个结果可以用除法法则
- 这三个导数都在公式的小册子
- 为线性函数
" class="Wirisformula" role="math" alt="加粗空格加粗斜体a加粗斜体x加粗斜体b" style="vertical-align:-6px;height:22px;width:55px">,在那里a x + b {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="加粗斜体a" style="vertical-align:-6px;height:22px;width:11px">而且a {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="粗体斜体b" style="vertical-align:-6px;height:22px;width:12px">是常数,b {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" alt="开始mathsize 16px样式加粗空格加粗斜体y加粗等于加粗sin加粗左括号加粗斜体a加粗斜体x加粗斜体b加粗右括号结束样式" style="vertical-align:-5px;height:20px;width:107px">是y = sin ( a x + b ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开始mathsize 16px样式空格分数分子粗体d粗体斜体y除以分母粗体d粗体斜体x结束分数粗体等于粗体斜体a粗体cos粗体左括号粗体斜体x粗体加粗体斜体b粗体右括号结束样式" style="vertical-align:-15px;height:41px;width:135px">d y d x = a cos ( a x + b ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" alt="开始mathsize 16px样式加粗空格加粗斜体y加粗等于加粗cos加粗左括号加粗斜体a加粗斜体x加粗斜体b加粗右括号结束样式" style="vertical-align:-5px;height:20px;width:110px">是y = cos ( a x + b ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开始mathsize 16px样式加粗空格分数分子加粗d加粗斜体y除以分母加粗d加粗斜体x结束分数加粗等于加粗减去加粗斜体a加粗sin加粗左括号加粗斜体a加粗斜体x加粗斜体b加粗右括号结束样式" style="vertical-align:-15px;height:41px;width:147px">d y d x = - a sin ( a x + b ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" alt="开始mathsize 16px样式加粗空格加粗斜体y加粗等于加粗tan加粗左括号加粗斜体a加粗斜体x加粗斜体b加粗右括号结束样式" style="vertical-align:-5px;height:20px;width:108px">是y = tan ( a x + b ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开始mathsize 16px样式粗体空格分数分子粗体d粗体斜体y /分母粗体d粗体斜体x结束分数粗体等于分数分子粗体a /分母粗体cos的粗体2的幂粗体空格粗体左括号粗体斜体a粗体斜体x粗体斜体加粗体斜体b粗体右括号结束分数结束样式" style="vertical-align:-17px;height:43px;width:146px">d y d x = a cos 2 ( a x + b ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 的导数
- 为一般函数
" class="Wirisformula" role="math" alt="加粗空格加粗斜体f加粗左括号加粗斜体x加粗右括号" style="vertical-align:-6px;height:22px;width:41px">,f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" style="vertical-align:-5px;height:20px;width:92px">是y = sin ( f ( x ) ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" style="vertical-align:-14px;height:41px;width:152px">d y d x = f ' ( x ) cos ( f x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" style="vertical-align:-5px;height:20px;width:95px">是y = cos ( f ( x ) ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" style="vertical-align:-14px;height:41px;width:163px">d y d x = - f ' ( x ) sin ( f ( x ) ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" alt="开始mathsize 16px样式加粗空格加粗斜体y加粗等于加粗tan加粗左括号加粗斜体f加粗左括号加粗斜体x加粗右括号加粗右括号结束样式" style="vertical-align:-5px;height:20px;width:97px">是y = tan ( f ( x ) ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开始mathsize 16px样式粗体空格分数分子粗体d粗体斜体y除以分母粗体d粗体斜体x结束分数粗体等于分数分子粗体斜体f粗体撇号粗体左括号粗体斜体x粗体右括号粗体右括号结束分数结束样式粗体2粗体空格粗体左括号粗体斜体f粗体左括号粗体斜体x粗体右括号粗体右括号结束分数结束样式" style="vertical-align:-17px;height:45px;width:136px">d y d x = f ' ( x ) cos 2 ( f ( x ) ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 的导数
- 后三个结果可以用链式法则
- 对于三角函数的微积分,角必须用弧度
- 确保你知道如何在GDC上改变角度模式
考试技巧
- 当你看到一个涉及到微分和三角函数的问题时,将你的GDC设置为弧度模式
工作的例子
" class="Wirisformula" role="math" alt="空间f左括号x右括号等于sinx" style="vertical-align:-6px;height:22px;width:88px" loading="lazy">f ( x ) = sin x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格f左括号x右括号等于cos左括号5x + 1右括号" style="vertical-align:-6px;height:22px;width:134px" loading="lazy">f ( x ) = cos ( 5 x + 1 ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
对e^x和lnx求导
如何区分指数和对数?
- 的导数
" class="Wirisformula" role="math" alt="加粗空格,加粗斜体y,加粗等于e的加粗斜体x次方" style="vertical-align:-6px;height:23px;width:52px" loading="lazy">是y = e x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开始mathsize 16px样式粗体空格分数分子粗体d粗体斜体y除以分母粗体d粗体斜体x结束分数粗体等于粗体e的粗体斜体x结束样式的幂" style="vertical-align:-15px;height:41px;width:65px" loading="lazy">在哪里d y d x = e x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="X元素的实数" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">x ∈ ℝ {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" alt="加粗空格加粗斜体y =加粗ln加粗空格加粗斜体x" style="vertical-align:-6px;height:22px;width:67px" loading="lazy">是y = ln x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开始mathsize 16px样式粗体空格分数分子粗体d粗体斜体y除以分母粗体d粗体斜体x结束分数粗体等于粗体1除以粗体斜体x结束样式" style="vertical-align:-15px;height:41px;width:67px" loading="lazy">在哪里d y d x = 1 x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格x大于0" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">x > 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 为线性函数
" class="Wirisformula" role="math" alt="加粗空格加粗斜体a加粗斜体x加粗斜体b" style="vertical-align:-6px;height:22px;width:55px" loading="lazy">,在那里a x + b {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="一个" style="vertical-align:-6px;height:22px;width:10px" loading="lazy">而且a {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="b" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">是常数,b {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" alt="加粗空格加粗斜体y加粗等于加粗e的左括号加粗斜体a加粗斜体x加粗斜体b加粗右括号结束指数" style="vertical-align:-6px;height:23px;width:89px" loading="lazy">是y = e ( a x + b ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" style="vertical-align:-15px;height:41px;width:109px" loading="lazy">d y d x = a e ( a x + b ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" style="vertical-align:-5px;height:20px;width:98px" loading="lazy">是y = ln ( a x + b ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" style="vertical-align:-15px;height:42px;width:108px" loading="lazy">d y d x = a ( a x + b ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 在特殊情况下
" class="Wirisformula" role="math" alt="空格b = 0" style="vertical-align:-6px;height:22px;width:41px" loading="lazy">,b = 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开始mathsize 16px样式粗体空格分数分子粗体d粗体斜体y除以分母粗体d粗体斜体x结束分数粗体等于粗体1除以粗体斜体x结束样式" style="vertical-align:-15px;height:41px;width:67px" loading="lazy">(d y d x = 1 x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="一个" style="vertical-align:-6px;height:22px;width:10px" loading="lazy">取消)a {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 在特殊情况下
- 的导数
- 为一般函数
" class="Wirisformula" role="math" alt="加粗空格加粗斜体f加粗左括号加粗斜体x加粗右括号" style="vertical-align:-6px;height:22px;width:41px" loading="lazy">,f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" style="vertical-align:-4px;height:25px;width:71px" loading="lazy">是y = e f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" style="vertical-align:-14px;height:41px;width:124px" loading="lazy">d y d x = f ' ( x ) e f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数
" class="Wirisformula" role="math" style="vertical-align:-5px;height:20px;width:89px" loading="lazy">是y = ln ( f ( x ) ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" style="vertical-align:-14px;height:41px;width:142px" loading="lazy">d y d x = f ' ( x ) ln ( f ( x ) ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 的导数
- 最后两组结果可以使用链式法则
考试技巧
- 记住要避免常见的错误:
- 的导数
" class="Wirisformula" role="math" alt="空间ln空间kx" style="vertical-align:-6px;height:22px;width:42px" loading="lazy">关于ln k x {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">是x {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="空间1 / x" style="vertical-align:-17px;height:47px;width:23px" loading="lazy">,不1 x {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="空间k / x" style="vertical-align:-17px;height:47px;width:23px" loading="lazy">k x {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} - 的导数
" class="Wirisformula" role="math" alt="就是e的k次方x端指数" style="vertical-align:-6px;height:23px;width:24px" loading="lazy">关于e k x {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">是x {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="K e的K次方x端指数" style="vertical-align:-6px;height:23px;width:34px" loading="lazy">,而不是k e k x {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="K x e的K (x - 1)次方的末端指数" style="vertical-align:-6px;height:23px;width:62px" loading="lazy">k x e k x - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"}
- 的导数
工作的例子
曲线有方程
求曲线在这一点的梯度