Activity Math 192

Definition of Derivative and Derivative of Polynomials

Derivative of Product and Quotient and Chain Rules

Derivative of Trigonometric Functions

Implicit Derivative and Logarithmic Functions

True-False

100

What is the definition of derivative?

f^'(x)= limit as h approaching 0 of [f(x+h) - f(x)]/h

100

Derivative of f(x)g(x)=?

Derivative of f(x)/g(x)=?

Derivative of f(x)g(x)=f^'(x)g(x)+f(x)g^'(x)

Derivative of f(x)/g(x)=

[g(x)f^'(x) - f(x)g^'(x)]/[(g(x))^2]

100

What is the derivative of tanx=?

What is the derivative of cotx=?

sec²x

-csc²x

100

Find dy/dx by implicit differentiation.

5x² - y³ = 7

y^'=10x/3y²

100

If y=e², then y^'=2e

False

200

How can a function fail to be differentiable?

Three ways for f not to be differentiable at a: a corner, a discontinuity, and a vertical tangent

200

Derivative of f(g(x))=? Find the derivative of f(x)=e^sqrt(x)

Derivative of f(g(x))=f^'(g(x)) g^'(x)

f^'(x)=(e^sqrt(x))(1/2sqrt(x))

200

Differentiate

f(x)=3sinx - 2cosx

f^'(x)=3cosx + 2sinx

200

d/dx (log_bx) = ?

1/(xlnb)

200

d/dx [sqrt(f(x))] = f^'(x)/2sqrt(f(x))

True

300

If f(x)=x², use the definition of derivative to find f'(x).

f^'(x)=2x

300

Differentiate

f(x)=(3x² -5) (e^x)

f^'(x)=(e^x)(3x² + 6x - 5)

300

Differentiate y=secx tanx

y^'=secx(sec²x+tan²x)

300

Find dy/dx by implicit differentiation.

sinx + cosy = 2x - 3y

y^'=(2y - x)/(y - 2x)

300

d/dx (10^x) = x 10^x-1

False

400

Find an equation of the tangent line to the curve at the given point.

f(x)=2x³ - x² + 2, (1,3)

y=4x - 1

400

Defferentiate

f(t)=5t/(t³ - t - 1)

f^'(t)=(-10t³ - 5)/(t³ - t - 1)³

400

If g(x)=(sinx)/x find g^'(x).

g^'(x)=(xcosx - sinx)/x²

400

Differentiate y=log₈(x²+3x)

(2x+3)/[(x²+3x)(ln8)]

400

d/dx (ln 10) = 1/10

False

500

If f(x)=sqrt(3-5x), use the definition of a derivative to find f^'(x).

f^'(x)=(-5/2)[(3-5x)^-1/2]

500

Find the derivative of the function. y=sqrt(x/(x+1))

y^'=1/[2(sqrt(x))(x+1)^3/2]

500

Find d/dx (arctanx) = ? Show your work.

Note: arctanx is the inverse tangent of x.

d/dx (arctanx) = 1/(1+x²)

500

If f(x) + x²[f(x)]³ = 10 and f(1) = 2. find f'(1).

-16/13

500

An equation of the tangent line to the parabola y=x² at (-2,4) is y - 4 = 2x (x +2).

True