Calculus I - Derivative Laws

17 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. Difference-Quotient definition of the derivative

   ddxf(x)=lim⁡h→0f(x+h)−f(x)h\frac{d}{dx}f(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}dxd​f(x)=limh→0​hf(x+h)−f(x)​

2. What is means to be differentiable at a point

   if f(x)f(x)f(x) is continuous at some point ccc , and the one-sided derivatives at point ccc agree (transition smoothly, like limits) then f′(x)f'(x)f′(x) exists at ccc , also stated as f(x)f(x)f(x) being differentiable at ccc .

3. Derivative notation

   f′(x)=ddxf(x)=dfdx=f'(x)=\frac{d}{dx}{f(x)}=\frac{df}{dx}=f′(x)=dxd​f(x)=dxdf​= the first derivative of x f′′(x)=d2dx2f(x)=d2fdx2=f''(x)=\frac{d^2}{dx^2}{f(x)}=\frac{d^2f}{dx^2}=f′′(x)=dx2d2​f(x)=dx2d2f​= the second derivative of x ... fn(x)=dndxnf(x)=dnfdxn=f^n(x)=\frac{d^n}{dx^n}{f(x)}=\frac{d^nf}{dx^n}=fn(x)=dxndn​f(x)=dxndnf​= the nth derivative of x

4. Derivative Evaluation notation

   f′(a)=ddxf(x)∣x=a=dfdx∣x=af'(a)=\frac{d}{dx}f(x)\Big|_{x=a}=\frac{df}{dx}\Big|_{x=a}f′(a)=dxd​f(x)∣ ∣​x=a​=dxdf​∣ ∣​x=a​ fn(a)=dndxnf(x)∣x=a=dnfdxn∣x=af^n(a)=\frac{d^n}{dx^n}f(x)\Big|_{x=a}=\frac{d^nf}{dx^n}\Big|_{x=a}fn(a)=dxndn​f(x)∣ ∣​x=a​=dxndnf​∣ ∣​x=a​

5. Constant rule

   If c∈Rc \in \mathbb{R}c∈R , then ddx(c)=0\frac{d}{dx}(c)=0dxd​(c)=0

6. Power rule

   If n∈Rn \in \mathbb{R}n∈R , then ddx(xn)=nxn−1\frac{d}{dx}(x^n)=nx^{n-1}dxd​(xn)=nxn−1

7. Constant multiple rule

   if c∈Rc \in \mathbb{R}c∈R , then ddx(cf(x))=cf′(x)\frac{d}{dx}(cf(x))=cf'(x)dxd​(cf(x))=cf′(x)

8. For all of the following assume the functions mentioned are differentiable at x

   Used in the following flashcards

9. Sum rule

   ddx(f(x)±g(x))=f′(x)±g′(x)\frac{d}{dx}(f(x) \pm g(x))=f'(x) \pm g'(x)dxd​(f(x)±g(x))=f′(x)±g′(x)

10. Product rule

    ddx(f(x)g(x))=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}(f(x)g(x))=f'(x)g(x)+f(x)g'(x)dxd​(f(x)g(x))=f′(x)g(x)+f(x)g′(x)

11. Quotient rule

    ddx(f(x)g(x))=f′(x)g(x)−f(x)g′(x)(g(x))2\frac{d}{dx}(\frac{f(x)}{g(x)})=\frac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2}dxd​(g(x)f(x)​)=(g(x))2f′(x)g(x)−f(x)g′(x)​

12. Chain rule

    ddx(f(g(x)))=f′(g(x))g′(x)\frac{d}{dx}(f(g(x)))=f'(g(x))g'(x) dxd​(f(g(x)))=f′(g(x))g′(x)

13. Implicit differentiation

    Used when it is impossible to solve for yyy explicitly, so we treat yyy as a function of xxx and apply the chain rule ddxy=y′dydx\frac{d}{dx}y=y'\frac{dy}{dx}dxd​y=y′dxdy​ Solve for dydx\frac{dy}{dx}dxdy​ to get the derivative

14. Normal line

    The line perpendicular to the tangent line (i.e. the derivative) at a given point

15. Logarithmic Differentiation

    If the function y=f(x)y=f(x)y=f(x) consists of products, quotients, and powers, take the natural logarithm of each side, then simplify using properties of logarithms Differentiate implicitly, and use the fact that ddxln⁡(y)=y′y\frac{d}{dx} \ln(y) = \frac{y'}{y}dxd​ln(y)=yy′​ Solve for y′y'y′ , and replace yyy with f(x)f(x)f(x)

16. Corner

    A non-differentiable point on a graph where derivatives exist on both sides of the point, but are not equal (Ex. ∣x∣|x|∣x∣ ).

17. Cusp

    A non-differentiable point on a graph which resembles a corner, yet features a slope that approaches some infinity ("infinitely sharp corners", ex. x23x^{\frac{2}{3}}x32​ )