衍生品概论
- 在介绍导数,一种理解一种限制是有益的
什么是极限?
- 的限制的函数值是(的函数吗
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">)方法如下x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">从任意一侧接近特定值x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 当函数在某一特定值处未定义时,极限就有意义了
- 例如,t他的功能
" class="Wirisformula" role="math" alt="空格f左括号x右括号等于分子x的4 - 1次方除以分母x - 1的末端分数" style="vertical-align:-17px;height:48px;width:104px" loading="lazy">会趋近于极限为f ( x ) = x 4 - 1 x - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">从下面和上面都接近1,但在没有定义x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="X = 1" style="vertical-align:-6px;height:22px;width:37px" loading="lazy">因为这需要除以0x = 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
关于极限我可能会被问到什么?
- 你可能会被要求从函数值表或的图中预测或估计极限
" class="Wirisformula" role="math" alt="Y = f左括号x右括号" style="vertical-align:-6px;height:22px;width:59px" loading="lazy">y = f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 你可能会被要求使用你的GDC绘制图表,并使用其中的值来估计极限
导数是什么?
- 微积分是关于利率的改变
- 汽车在道路上位置的变化方式是它的速度(速度)
- 汽车速度变化的方式是它的加速度
- 的梯度(非线性)的(变化率)函数随
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数函数的是一个函数,它与梯度的价值
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 导数也叫做梯度函数
极限和导数是如何联系在一起的?
- 考虑一下这一点
" class="Wirisformula" role="math" alt="P" style="vertical-align:-6px;height:22px;width:12px" loading="lazy">在图上P {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Y = f左括号x右括号" style="vertical-align:-6px;height:22px;width:59px" loading="lazy">如下图所示y = f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="左方括号P Q下标i右方括号" style="vertical-align:-12px;height:28px;width:42px" loading="lazy">是一连串的和弦吗[ P Q i ] {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 的梯度的函数
" class="Wirisformula" role="math" alt="空格f左括号x右括号" style="vertical-align:-6px;height:22px;width:36px" loading="lazy">在这一点上f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="P" style="vertical-align:-6px;height:22px;width:12px" loading="lazy">是平等的到梯度的切点P {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="P" style="vertical-align:-6px;height:22px;width:12px" loading="lazy">P {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的梯度的切点
" class="Wirisformula" role="math" alt="P" style="vertical-align:-6px;height:22px;width:12px" loading="lazy">是限制的梯度和弦的声音P {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="左方括号P Q下标i右方括号" style="vertical-align:-12px;height:28px;width:42px" loading="lazy">作为点[ P Q i ] {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="问" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">“滑”下曲线越来越接近目标Q {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="P" style="vertical-align:-6px;height:22px;width:12px" loading="lazy">P {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的梯度函数的变化
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">变化x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的导数函数是根据值计算梯度的吗
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
导数的符号是什么?
- 对于函数
" class="Wirisformula" role="math" alt="Y = f左括号x右括号" style="vertical-align:-6px;height:22px;width:59px" loading="lazy">,的导数,关于y = f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">,可以写成x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 可以使用不同的变量
- 例如,如果
" class="Wirisformula" role="math" alt="V等于f左括号s右括号" style="vertical-align:-6px;height:22px;width:61px" loading="lazy">然后V = f ( s ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="分数分子V除以分母d s结束分数等于f撇号左括号s右括号" style="vertical-align:-17px;height:47px;width:83px" loading="lazy">d V d s = f ' ( s ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 例如,如果
关于衍生品我可能会被问到什么?
- 你可能会被要求使用GDC的绘图功能来查找梯度的函数在不同的值
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 从一个系列的梯度价值观,你可能会被要求提出一个表达式为导数(梯度函数)函数的
工作的例子
的图形
求x的幂的微分
什么是差异化?
- 分化这个过程是求的表达式吗导数(梯度函数)从一个函数的表达式
如何求导x的幂函数?
- 权力的
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">是有区别的根据以下公式:x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 如果
" class="Wirisformula" role="math" alt="空格f左括号x右括号等于x的n次方" style="vertical-align:-6px;height:23px;width:71px" loading="lazy">然后f ( x ) = x n {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格f撇号左括号x右括号等于nx的n - 1次幂的结束指数" style="vertical-align:-6px;height:24px;width:104px" loading="lazy">在哪里f ' ( x ) = n x n - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="N个有理数元素" style="vertical-align:-6px;height:22px;width:42px" loading="lazy">n ∈ ℚ {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 这是已知的公式的小册子
- 如果
- 如果
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">是增加由一个常数然后导数也乘以这个常数x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 如果
" class="Wirisformula" role="math" alt="空格f左括号x右括号等于ax的n次方" style="vertical-align:-6px;height:23px;width:79px" loading="lazy">然后f ( x ) = a x n {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格f撇号左括号x右括号等于anx的n - 1次方末端指数" style="vertical-align:-6px;height:24px;width:112px" loading="lazy">在哪里f ' ( x ) = a n x n - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="N个有理数元素" style="vertical-align:-6px;height:22px;width:42px" loading="lazy">而且n ∈ ℚ {"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">是常数a {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 如果
- 的替代符号(
" class="Wirisformula" role="math" alt="空格f撇号左括号x右括号" style="vertical-align:-6px;height:24px;width:41px" loading="lazy">)是使用f ' ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="分子dy除以分母dx,结束分数" style="vertical-align:-17px;height:47px;width:28px" loading="lazy">d y d x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 如果
" class="Wirisformula" role="math" alt="Y = ax的n次方" style="vertical-align:-6px;height:23px;width:54px" loading="lazy">然后y = a x n {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="分子dy除以分母dx末端分数等于anx的n - 1次方末端指数" style="vertical-align:-17px;height:47px;width:99px" loading="lazy">d y d x = a n x n - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 例如,如果
" class="Wirisformula" role="math" alt="Y = - 4x的1 / 2次方" style="vertical-align:-6px;height:42px;width:77px" loading="lazy">然后y = - 4 x 1 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="分数分子dy除以分母dx端分数等于- 4叉乘1 / 2叉乘x的1 / 2次方- 1端指数等于- 2x的- 1 / 2次方端指数" style="vertical-align:-17px;height:53px;width:240px" loading="lazy">d y d x = - 4 × 1 2 × x 1 2 - 1 = - 2 x - 1 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 例如,如果
- 如果
- 不要忘记这些两个特殊情况:
- 如果
" class="Wirisformula" role="math" alt="空格f左括号x右括号等于ax" style="vertical-align:-6px;height:22px;width:71px" loading="lazy">然后f ( x ) = a x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格f撇号左括号x右括号等于a" style="vertical-align:-6px;height:24px;width:67px" loading="lazy">f ' ( x ) = a {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 例如,如果
" class="Wirisformula" role="math" alt="Y = 6x" style="vertical-align:-6px;height:22px;width:47px" loading="lazy">然后y = 6 x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="分子dy除以分母dx端分数等于6" style="vertical-align:-17px;height:47px;width:54px" loading="lazy">d y d x = 6 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 例如,如果
- 如果
" class="Wirisformula" role="math" alt="空格f左括号x右括号等于a" style="vertical-align:-6px;height:22px;width:62px" loading="lazy">然后f ( x ) = a {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格f撇号左括号x右括号等于0" style="vertical-align:-6px;height:24px;width:67px" loading="lazy">f ' ( x ) = 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 例如,如果
" class="Wirisformula" role="math" alt="Y = 5" style="vertical-align:-6px;height:22px;width:37px" loading="lazy">然后y = 5 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="分子dy除以分母dx,结束分数等于0" style="vertical-align:-17px;height:47px;width:54px" loading="lazy">d y d x = 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 例如,如果
- 这些可以让你求导线性条件在
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">而且常量x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 如果
- 功能涉及根将需要重写为部分权力的
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">第一个x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 例如,如果
" class="Wirisformula" role="math" alt="空格f左括号x右括号等于2√x" style="vertical-align:-6px;height:26px;width:90px" loading="lazy">然后重写为f ( x ) = 2 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2x的1 / 2次幂" style="vertical-align:-6px;height:42px;width:85px" loading="lazy">和区分f ( x ) = 2 x 1 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 例如,如果
- 功能涉及分数与分母在这方面
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">将需要重写为负面的力量的x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">第一个x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 例如,如果
" class="Wirisformula" role="math" alt="空格f左括号x右括号= 4 / x" style="vertical-align:-17px;height:47px;width:71px" loading="lazy">然后重写为f ( x ) = 4 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4x的- 1次幂" style="vertical-align:-6px;height:23px;width:89px" loading="lazy">和区分f ( x ) = 4 x - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 例如,如果
如何求导x的和和和幂的差呢?
- 幂的微分公式
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">适用于所有理性的幂,所以有可能微分任何表达式是a总和或区别的权力的x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 例如,如果
" class="Wirisformula" role="math" alt="空格f左括号x右括号等于5x的4次方- 3x的2 / 3次方端指数加4" style="vertical-align:-6px;height:42px;width:154px" loading="lazy">然后f ( x ) = 5 x 4 - 3 x 2 3 + 4 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格f撇号左括号x右括号等于5叉乘乘以4x的4次方- 1端指数减去3叉乘乘以2 / 3x的2次方/ 3 - 1端指数加0" style="vertical-align:-17px;height:53px;width:257px" loading="lazy">f ' ( x ) = 5 × 4 x 4 - 1 - 3 × 2 3 x 2 3 - 1 + 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格f撇号左括号x右括号等于20x³- 2 x的- 1次方的第三次方端指数" style="vertical-align:-6px;height:42px;width:152px" loading="lazy">f ' ( x ) = 20 x 3 - 2 x - 1 3 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 例如,如果
- 产品而且商不能以这种方式进行区分是需要的扩大/简化第一个
- 例如,如果
" class="Wirisformula" role="math" alt="空格f左括号x右括号等于左括号2x - 3右括号左括号x²- 4右括号" style="vertical-align:-6px;height:23px;width:165px" loading="lazy">然后展开到f ( x ) = ( 2 x - 3 ) ( x 2 - 4 ) {"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2x³- 3x²- 8x + 12" style="vertical-align:-6px;height:23px;width:193px" loading="lazy">哪一个是一个和/差的幂f ( x ) = 2 x 3 - 3 x 2 - 8 x + 12 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">可以被区分x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 例如,如果
一旦我找到了导数(梯度函数),我可能会被要求做什么?
- 的导数可以用来找梯度的函数在任何时候
- 函数在一点处的梯度等于梯度的切这一点的曲线
- 一个问题可能指的是切线的梯度
- GDC可用于帮助寻找梯度或验证答案
- 的负次幂和分数次幂微分时要格外小心
" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="1 / 2 - 1 = - 1 / 2" style="vertical-align:-17px;height:47px;width:95px" loading="lazy">但1 2 - 1 = - 1 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="1 / 3 - 1 = - 2 / 3" style="vertical-align:-17px;height:47px;width:95px" loading="lazy">1 3 - 1 = - 2 3 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
考试技巧
- 一个常见的错误是在求导之前不化简表达式
- 的导数
" class="Wirisformula" role="math" alt="左括号(x²+ 3)右括号(x³- 2x + 1)右括号" style="vertical-align:-6px;height:23px;width:147px" loading="lazy">可以不的导数相乘就可以得到x 2 + 3 x 3 - 2 x + 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="开括号x²+ 3个闭括号" style="vertical-align:-6px;height:23px;width:56px" loading="lazy">而且x 2 + 3 {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="左括号是(x³- 2x + 1)右括号是" style="vertical-align:-6px;height:23px;width:92px" loading="lazy">x 3 - 2 x + 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"}
- 的导数
工作的例子
这个函数
在哪里