MCQ 2

Product Rule with Trig Tips

- To keep the radical in the equation when finding the derivative, put a one on top, the 2 on bottom and move the radical to the bottom ex. square root of 2 = 1/2srx ex. Correct. The product rule is applied, as follows: f1(x)=d/dx(srx)cos⁡x+srxd/dx(cos⁡x)=1/2srx times cos⁡x+srx(−sin⁡x)=cos⁡x−2x⁢sin⁡x/2srx.

Finding the derivative of the function using the QUOTIENT RULE TIPS

- first, find the derivative on the top - next, fill in the given number into the equation (top and bottom; do not find the derivative of the bottom) - solve!

d/dx cotx

-csc^2x

d/dx cscx

-cscxcotx

d/dx cosx

-sinx

Given the function (f), where b is a constant and g is a differentiable function satisfying g(a)=__ and g1(a)=__. For what value of b is f1(a) =b?

CHECK TO SEE IF YOU ARE MULTIPLYING< DIVIDING< ADDING< SUBTRACTING> ETC> Use the product rule to find f1(x). Then evaluate x=a and set equal to b. ex. f(x)=(x^3+bx+6)g(x) ...... g(2)=3, g1(2)=-1, what value of b is f1(2)=0 f′(x)=(3x2+b)g(x)+(x3+bx+6)g′(x)f′(x)=(3x2+b)g(x)+(x3+bx+6)g′(x) f′(2)=(12+b)(3)+(8+2b+6)(−1)=22+b=0⇒b=−22f′(2)=(12+b)(3)+(8+2b+6)(−1)=22+b=0⇒b=−22

Given the function, what is the instantaneous rate of change of the function with respect to x at x=____ ?

Equation is in the lim (as h-0) f(x+h) - f(x) / h Take the limit of the difference quotient - Solve the equation all the way. Then insert the value into the equation and solve. ex. g(x)=lim(h-0) sin(x+h)-sin(x) / h at x=pi/3 g(x)=sin The derivative of sin is -cos, but we need the answer in sin. FInd the answer using -cos but then change back to sin---- sin(pi/3)= -rad3/2 (may need unit circle)

What is the slope of the tangent line at a specific point?

Find the derivative of the equation (check to see what rule you need). Then plug in the x value into the equation and solve.

Given the function, find the derivative of the function of (_____)

First, find the derivative of the function. Then insert the number given into the equation in the spot of the x value. ex. f(x)= x^3+3x^2-4 find f1(2)

GIven a function, find all values of x that the derivative of the function equals _____.

This requires a calculator.... but here's the shortcut Find the derivative of the function using the power rule. Then use guess and check with the answers given to see what equals the number given. ex. g(x)= x^4-3x^3-x that g1(x)=1/2

Using the table, what is d/dx(f(x)g(x)) at x=4?

USE THE PRODUCT RULE!!!! f1(x)g(x) + g1(x)f(x) and solve.

The graphs of f and g are shown. Find h1(a)= ?

Use the equation of h(x) to find the derivative of h(x). Then insert the value a into the equation Find the value of f1(3), f(3), g1(3), and g(3) on the graph Insert these values into the equation and solve.

Finding the instant rate of change

Use the formula lim (as h-0) f(x+h) -f(x) /h ex. t. lim(h→0)5e^x−5e^x+h/3h=5/3lim(h→0)e^x−e^x+h/h=−5/3lim(h→0)e^x+h−e^x/h=−5/3(d/dx e^x) =−5/3e^x

d/dx sinx

cosx

d/dx tanx

sec^2x

d/dx secx

secxtanx