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Sum and Difference Rules

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Jan 24, 2022

## Cards(30)

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(30 cards)

Sum and Difference Rules

Front

The sum/difference of two differentiable function is differentiable and is the sum/difference of their derivatives

$$\frac{d}{dx}[f(x)+g(x)]=f'(x)+g'(x)$$ sum rule

$$\frac{d}{dx}[f(x)-g(x)]=f'(x)-g'(x)$$ difference rule

Back

$$\frac{d}{dx}[e^x]$$

Front

$$e^x$$

Back

Approximate $$f'(c)$$

Front

Slope of secant:

$$f'(c)\approx\frac{f(a)-f(b)}{a-b}$$

Back

Find the Derivative of $$f(x)$$ at $$x=c$$

Front

Find $$f'(c)$$

Back

$$\frac{d}{dx}[\frac{f(x)}{g(x)}]$$

Front

$$\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$$

Back

Equation of a tangent line

If $$m_{\tan}$$ exists, then the equation of the tangent line to the graph of $$f$$ at the point $$(c, f(c))$$ is...

Front

$$y-f(c)=f'(c)(x-c)$$

Back

Tangent Line

Front

$$\lim_{x\to c} \frac{f(x)-f(c)}{x-c}$$

Back

Slope of Vertical Tangent

Front

$$f'(c)\to\text{undefined slope}$$

Back

$$m_{secant}$$

Front

$$\frac{f(x)-f(c)} {x-c}$$

Back

Find the Average Velocity/ROC at $$x=c$$

Front

Slope of secant:

$$f'(c)=\frac{f(a)-f(b)}{a-b}$$

Back

Derivatives that fail to exist

Front

Corner

Sharp Turn (Cusp)

Vertical Tangent

Back

True or False, if $$f$$ is continuous at $$x=c$$, then $$f$$ is differentiable at $$x=c$$

Front

False

Back

Definition of the Derivative Function $$f'(x)$$

Front

$$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$

Back

Vertical Tangent Line

Front

If $$f$$ is continuous at $$x=c$$, and $$\lim_{h\to 0}\frac{f(c+h)-f(c)}{h} = \pm\infty$$, then the vertical line passing through $$(c, f(c))$$ is a vertical tangent line to the graph of $$f(x)$$

Back

Equation of Tangent Line

Front

POT: Point of tangent, $$f(x)$$

SOT: Slope of tangent, $$f'(x)$$

Back

Another way to write $$h\to 0$$

Front

$$\Delta x\to 0$$

Back

$$\frac{d}{dx}[\ln x]$$

Front

$$\frac{1}{x}$$

Back

$$\frac{d}{dx}[f(x)\cdot g(x)]$$

Front

$$f'(x)\cdot g(x) + f(x)\cdot g'(x)$$

Back

Constant Multiple Rule

Front

$$\frac{d}{dx}[c\cdot f(x)]=c\cdot f'(x)$$

Back

$$\frac{d}{dx}[\sin x]$$

$$\frac{d}{dx}[\cos x]$$

Front

$$\cos x$$

$$-\sin x$$

Back

$$\frac{d}{dx}[\log_bx]$$

Front

$$\frac{1}{(\ln b)x}$$

Back

Slope of horizontal tangent

Front

$$f'(c)=0$$

Back

Trig Identities

Front

$$\sin^2x+\cos^2x=1$$

$$1+\cot^2x=\csc^2x$$

$$\tan^2x+1=\sec^2x$$

Back

Find the slope of the secant line at $$x=c$$

Front

Slope of secant:

$$f'(c)\approx\frac{f(a)-f(b)}{a-b}$$

Back

Find the Instantaneous Velocity or ROC at $$x=c$$

Front

Find $$f'(c)$$

Back

Definition of Derivative at $$x=c$$

Front

$$\lim_{x\to c}\frac{f(x)-f(c)}{x-c}$$ or $$\lim_{h\to 0}\frac{f(c+h)-f(c)}{h}$$

Back

Point slope formula

Front

$$y-y_1=m(x-x_1)$$

Back

Derivatives of $$\sec, \tan, \csc, \cot$$

Front

$$\sec\rightarrow\sec\leftarrow\tan$$

$$\csc\rightarrow-\csc\leftarrow\cot$$

Back

Find the slope of the tangent line (or graph) of $$f(x)$$ at $$x=c$$

Front

Find $$f'(c)$$

Back

True or False, if $$f$$ is differentiable at $$x-c$$, then $$f$$ is continuous at $$x=c$$

Front

True

Back