Intro to Derivative Calculus

A Slippery Slope

100

This calculus tool is related to the concept of slope at a point

What is the derivative?

200

This formula defines the average rate of change of a function

{f(b)-f(a)}/{b-a}

300

This is the formal definition of the derivative using limits

f'(x)=lim_{h->0}{f(x+h)-f(x)}/h

400

The derivative of what function is represented in this limit: 

lim_{h->0}{(x+h)^4*sin(x+h)-(x^4)*sin(x)}/h

y=x^4*sin(x)

500

What is the derivative of this function at x=0: 

lim_{h->0}{sec(x+h)-sec(x)}/h

1

Elementary,My Dear Function

100

d/{dx}e^x

e^x

200

d/{dx}x^n

n*x^{n-1}

300

d/{dx}\log_a(x)

1/{x\cdot\ln(a)}

400

d/{dx}(f(x)\cdot g(x))

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

500

d/dx (f(x)/g(x))

(g(x)*f'(x)-f(x)*g'(x))/(g(x))^2

I've Got the Power!

100

f(x)=x

f'(x)=1

200

f(x)=x^6-2x^3

f'(x)=6x^5-6x^2

300

f(x)=4x-\sqrt{x}

f'(x)=4-1/{2\sqrt{x}}

400

f(x)=root(6){x^5}+1/{x^2}

f'(x)=5/{6root(6){x}}-2/{x^3}

500

f(x)=x^4*\sin(x)+\sqrt{x^3}*e^x

f(x)=x^4\cos(x)+4x^3\sin(x)+\sqrt{x^3}*e^x+{3\sqrt{x}}/2*e^x}

What's My Line, Again?

100

Find the equation of the tangent line at x=3 of 

f(x)=5x^2

y-45=30(x-3)

200

Find the equation of the tangent line at x=0 of

f(x)=e^x

y-1=x

300

Find the equation of the tangent line at

x=\pi/4

f(x)=cos(x)

y-\sqrt{2}/2=-\sqrt{2}/2(x-\pi/4)

400

Find the equation of the normal line at x=0 of

y=e^x*\tan(x)

y=-x

500

Find the equation of the normal line at x=3 of 

y=\ln(x)/{x^2}

y-\ln(3)/9=-{27}/{1-2\ln(6)}(x-3)