BAS CAL | Derivatives
Tangent and Secant Lines
Tangent Line
a line that is tangent to a curve or a part of a function at only one point
Secant Line
a line that crosses a curve or a part of a function at two points
Tangent and Secant line
To create the tangent line, the two points of the secant line must approach each other, justifying the use of limits.
How to solve for slope of tangent line
Formula
m_{\sec}=\frac{y_2-y_1}{x_2-x_1}
m_{\tan}=\lim_{x_2\to x_1}\frac{y_2-y_1}{x_2-x_1}
Steps
Let (x1, y1) be the given coordinates, and let (x2, y2) be arbitrary coordinates.
Replace (x1, y1) in the limit formula with the given coordinates.
Replace y2 with the function equation. You should end up with an indeterminate function or 0/0 when x is substituted with the given.
Solve the limit as usual.
Derivatives (f’)
Definition
“slope of the tangent line”
Forms
f^{\prime}\left(x\right) | “f prime of x”
y^{\prime} | “y prime”
dy/dx | “the derivative of y with respect to x
Derivatives in Relation to the Slope
Given points \left(x_0,y_0\right) ,
f^{\prime}\left(x_0\right)=\lim_{x\to x_0}\frac{f\left(x\right)-f\left(x_0\right)}{x-x_0}
D_{x}\left\lbrack f\left(x\right)\right\rbrack= d/dx[f(x)] = d/dx[y] = dy/dx
Formula (Delta Method)
f^{\prime}\left(x\right)=\lim_{x\to}\frac{f\left(x\right)-f\left(x+\Delta x\right)}{\Delta x}
Differentiation
the process of computing for the derivative
Main Derivative Rules
The Constant Rule
f^{\prime}\left(c\right)=0 | c is a constant
The Power Rule
f^{\prime}\left(x^{n}\right)=nx^{n-1} | n is a real number
The Constant Multiple
f^{\prime}\left(cx\right)=c\left(f^{\prime}\left(x\right)\right) | c is a constant number
The Sum and Difference Rule
f^{\prime}\left(g\left(x\right)\pm h\left(x\right)\right)=g^{\prime}\left(x\right)\pm h^{\prime}\left(x\right)
The Product Rule
f^{\prime}\left(g\left(x\right)\cdot h\left(x\right)\right)=g^{\prime}\left(x\right)\cdot h\left(x\right)+g\left(x\right)\cdot h^{\prime}\left(x\right)
The Quotient Rule
f^{\prime}\left(\frac{g\left(x\right)}{h\left(x\right)}\right)=\frac{h\left(x\right)g^{\prime}\left(x\right)-h^{\prime}\left(x\right)g\left(x\right)}{\left\lbrack g\left(x\right)\right\rbrack^2}
f^{\prime}\left(\frac{hi}{lo}\right)=\frac{lo\cdot d\left\lbrack hi\right\rbrack-hi\cdot d\left\lbrack lo\right\rbrack}{lolo}
The Chain Rule
f^{\prime}\left(h\left(g\left(x\right)\right)\right)=h^{\prime}\left(g\left(x\right)\right)\cdot g^{\prime}\left(x\right)
The General Power Rule
f^{\prime}\left(\left\lbrack g\left(x\right)\right\rbrack^{n}\right)=n\left\lbrack g\left(x\right)\right\rbrack^{n-1}\cdot g^{\prime}\left(x\right)
Rewriting Exponents
If a function or term is in the numerator, make its exponent negative
If a function or term is under an nth root, make its exponent 1/n
Other Derivative Rules
Exponential Functions
f^{\prime}\left(b^{x}\right)=b^{x}\ln b
f^{\prime}\left(e^{x}\right)=e^{x}
Logarithmic Functions
f^{\prime}\left(\log_{b}x\right)=\frac{1}{x\ln b}
f^{\prime}\left(\ln b\right)=\frac{1}{x}
Trigonometric Functions
f^{\prime}\left(\sin x\right)=\cos x
f^{\prime}\left(\cos x\right)=-\sin x
f^{\prime}\left(\tan x\right)=\sec^2x
f^{\prime}\left(\csc x\right)=-\csc x\cot x
f^{\prime}\left(\sec x\right)=\sec x\tan x
f^{\prime}\left(\cot x\right)=-\csc^2x
Normal Line
the line perpendicular to the tangent line
Higher Order Derivatives
repeated process of taking derivatives of derivatives