calc 2

Sum and Difference Rules

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

\(\frac{d}{dx}[e^x]\)

\(e^x\)

Approximate \(f'(c)\)

Slope of secant:

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

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

Find \(f'(c)\)

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

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

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...

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

Tangent Line

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

Slope of Vertical Tangent

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

\(m_{secant}\)

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

Find the Average Velocity/ROC at \(x=c\)

Slope of secant:

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

Derivatives that fail to exist

Corner

Sharp Turn (Cusp)

Vertical Tangent

True or False, if \(f\) is continuous at \(x=c\), then \(f\) is differentiable at \(x=c\)

False

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

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

Vertical Tangent Line

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 \(f(x)\)

Equation of Tangent Line

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

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

Another way to write \(h\to 0\)

\(\Delta x\to 0\)

\(\frac{d}{dx}[\ln x]\)

\(\frac{1}{x}\)

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

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

Constant Multiple Rule

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

\(\frac{d}{dx}[\sin x]\)

\(\frac{d}{dx}[\cos x]\)

\(\cos x\)

\(-\sin x\)

\(\frac{d}{dx}[\log_bx]\)

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

Slope of horizontal tangent

\(f'(c)=0\)

Trig Identities

\(\sin^2x+\cos^2x=1\)

\(1+\cot^2x=\csc^2x\)

\(\tan^2x+1=\sec^2x\)

Find the slope of the secant line at \(x=c\)

Slope of secant:

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

Find the Instantaneous Velocity or ROC at \(x=c\)

Find \(f'(c)\)

Definition of Derivative at \(x=c\)

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

Point slope formula

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

Derivatives of \(\sec, \tan, \csc, \cot\)

\(\sec\rightarrow\sec\leftarrow\tan\)

\(\csc\rightarrow-\csc\leftarrow\cot\)

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

Find \(f'(c)\)

True or False, if \(f\) is differentiable at \(x-c\), then \(f\) is continuous at \(x=c\)

True