Calculus Derivatives

Trig Derivatives

100

What is the derivative of y = sin(x) ?

y' = cos(x)

200

Find the derivative of f(x) = sec(x).

f'(x)=sec(x)tan(x)

300

Find the derivative of f(x) = csc²(2x)

-4((csc²(2x)cot(2x))

2(csc(2x)(-csc(2x)cot(2x))*2

400

Find the derivative of f(x) = cos(tan(x))

-sin(tan(x))sec²(x)

500

Find the derivative of y=tan^1/2(2x)

y'= (sec²(2x))/tan^1/2(2x)

Product and Quotient Rule

100

Find the derivative of y = 3x² sin x

3x² cos x + 6x sin x

200

f(x) = (3x - 2x²)(5 + 4x)

-24x² + 4x + 15

300

Find the derivative of y = (5x - 2)/(x² + 1)

(-5x² + 4x + 5)/(x² + 1)²

400

Find the derivative of f(x) = (2x + 5)/(x)^1/2

(2x - 5)/(2x^3/2)

(2x^1/2-((2x+5)/(2x^1/2))/x

500

Find f'(x) when f(x)= (x+4)/((x²+1)^1/2))

[((x²+1)^1/2 - ((x(x+4))/((x²+1)^1/2)]/[x²+1]

Chain Rule

100

Write the formula for chain rule.

d/dx[f(g(x))] = f'(g(x))*g'(x)

dy/dx= dy/du*du/dx

200

Find y' if y = (3x - 9)⁴

12(3x - 9)³

300

Find the derivative of f(x) = sin 4x

4cos(4x)

400

Find y' if y = (1+x^1/2)^1/2

(4x^1/2(1+x^1/2)^1/2))^-1

1/(4x^1/2(1+x^1/2)^1/2))

500

Find f''(x) if f(x) = (9x²+4)^1/3

(6(-3x²+4))/(9x²+4)^5/3

Inverses/Random

100

Find g'(x) if f(g(x))=x.

g'(x)=1/(f'(g(x)))

200

What is the derivative of csc(x)?

-csc(x)cot(x)

300

Write the limit definition of a derivative.

f'(x)=limh→0(f(x+h)−f(x))/h

400

If g(x) and f(x) are inverses and f(x)=x², find g'(4).(Hint: When is x²=4?)

g'(4)= 1/4

500

Write the equation of the tangent line of f^-1(x) at x=10 given f(x)=x³+7x+2 and f(1)=10.

y-1=1/10(x-10)

Calculator Questions

100

Find the average rate of change of f(x)=ln(3x) over the interval [1,4]

approximately 0.462

200

If f(x)=sin(x^1/2), find f'(2).

approximately 0.055

300

If f(x)=ln(1/(5-x)), find f'(1.3)

0.270

400

Write the equation of the line tangent to y=(x/(x³+1))^1/2 at x=1.

y-0.707=-0.1767(x-1)

500

f(x)=-2x/3 + 25/3 & g(x)=(x²-x+3)^1/2. m(x) is defined by m(x)=f(x)/2g(x). Find m'(5).

m'(x)= [(f'(x)*2g(x))-(f(x)*2g'(x))]/(2g(x))²

f'(5)=-2/3

g'(5)=0.938315

m'(5)=-0.171