•csch(x)—hyperbolic cosecant •arsinh(x)—inverse hyperbolic sine •arcosh(x)—inverse hyperbolic cosine •artanh(x)—inverse hyperbolic tangent •arcoth(x)—inverse hyperbolic cotangent •sec(x)—secant •csc(x)—cosecant
Inverse Trigonometry sin-1(x) 1/√(1−x2) cos-1(x) −1/√(1−x2) tan-1(x) 1/(1+x2) RulesFunction Derivative Multiplication by constant cf cf’ Power Rule xn nxn−1 Sum Rule f + g f’ + g’ Difference Rule f − g f’ − g’ Product Rule fg f g’ + f...
As the name suggests, anti-derivative is the inverse process of differentiation. The derivative of cos x is -sin x and the derivative of sin x is cos x. So, the anti-derivative of cos x is sin x + C and the anti-derivative of sin x is -cos x + C, where C is constant of ...
Derivative $f’$ of function $f(x)=\arccos{x}$ is: \[\forall x \in ]–1, 1[ ,\quad f'(x) = -\frac{1}{\sqrt{1-x^2}}\] Proof Remember that function $\arcsin$ is the inverse function of $\cos$ : \[\left(f^{-1} \circ f\right)=\left(\cos \circ \arccos\right)(...
, the ‘red’ P is the inverse of the 1st arrow Michael Penn’s Linear Algebra trick can be used for cases when finding Nth power of an expression (nth derivative, nth power polynomial…) : 1) create a square matrix (here he used a companion cos3x of sin3x ...
Derivative $f’$ of the function $f(x)=\sin x$ is: \(\forall x \in ]-\infty, +\infty[ , f'(x) = \cos x\) Proof/Demonstration \[\begin{aligned} \frac{\sin (x+h)-\sin x}{h}&= \frac{\sin (x) \cos (h)+\cos (x) \sin (h)-\sin x}{h} \\ \frac{\sin (x...
Thus: \(\begin{aligned} (\tan x)' &=\lim _{h \rightarrow 0} \frac{\tan (x+h) - \tan x }{h}\\ &=\lim _{h \rightarrow 0} \frac{\sin h}{h} \cdot \frac{1}{\cos h} \cdot \frac{1+\tan ^{2} x}{1-\tan x \tan h}\\ &=\left(\lim _{h \rightarrow 0} ...
Example: Compute the first derivative off(x)=sin(x)⋅ex. Using the product rule:f′(x)=ddx[sin(x)]⋅ex+sin(x)⋅ddx[ex]=cos(x)⋅ex+sin(x)⋅ex=ex(cos(x)+sin(x)) 2. Partial Derivative
dy/dx = 1 / (1 + x2) (because tan y = x) In this way, the implicit differentiation process can be used to find the derivatives of any inverse function. Important Notes on Implicit Differentiation: Implicit differentiation is the process of finding dy/dx when the function is of the for...
Inverse hyperbolic cosine cosh-1x Inverse hyperbolic tangent tanh-1x Derivative examples Example #1 f(x) =x3+5x2+x+8 f '(x) = 3x2+2⋅5x+1+0 = 3x2+10x+1 Example #2 f(x) = sin(3x2) When applying the chain rule: f '(x) = cos(3x2) ⋅ [3x2]' = cos(3x2) ⋅ 6x...