Derivative of Trigonometric Functions

Derivatives of Trig Functions

First....a limit you need to know.

The limit of of cos h minus 1 all divided by h as h goes to zero is equal to what? First, write this down in limit notation. Then watch the following video........

Now lets find f'(x) when f(x) = sin x ......

• Apply the definition of derivative.

• Use the addition of angles identity on sin(x + h).

• Now break up the limit into two. Lim A + Lim B. Use the limit I just taught you plus a familiar one to evaluate.

• You should end up with sin x * 0 + cos x * 1. Did you?

Therefore.....

• If f(x) = sin x, then f'(x) = cos x

• By the same method, d(cos x)/dx = - sin x

• NOW calculate d(tan x)/dx using these and the quotient rule.

Multiple Choice

The first derivative of tan x = ?

• sec x tan x

• - csc² x

• sec² x

• -csc x cot x

Multiple Choice

When we "take the derivative" of a function what are we finding?

• What's a derivative?

• The rate at which our struggles in Calculus are increasing.

• The slope of the secant line

• The slope of the tangent line

Multiple Choice

What is the derivative of cot(x)?

• sec²(x)

• -sec²(x)

• csc²(x)

• -csc²(x)

Multiple Choice

What is the derivative of sec(x)?

• sec(x)tan(x)
• csc(x)cot(x)
• -sec(x)tan(x)
• -csc(x)cot(x)

Multiple Choice

Find the derivative.

• x⁴ cos(x) - 4x³ sin(x)

• x⁴ cos(x) + 4x³ sin(x)

• 4x³ cos(x)

• -4x³ cos(x)

Multiple Choice

Find the derivative

$$f\left(x\right)=\frac{x^2}{\sin x}$$  

• $$\frac{2x\sin x-x^2\cos x}{\sin^2x}$$  

• $$\frac{2x\cos x-x^2\cos x}{\sin^2x}$$  

• $$\frac{2x\sin x+x^2\cos x}{\sin^2x}$$  

• $$\frac{2x\cos x-x^2\sin x}{\sin^2x}$$  

Multiple Choice

The velocity of an object is given as v = 2t + 3t³. what is the acceleration of the object at t = 2 s?

• 38 m/s/s

• 27 m/s/s

• 16 m/s/s

• 49 m/s/s

Multiple Choice

At which x-value is f continuous but not differentiable?

• a

• b

• c

• d

Multiple Choice

Set up the derivative of y=(3x⁴ - 7)*(5x² + 1)

• (12x³)*(10x)

• (3x⁴ - 7)*(10x) + (12x³)*(5x² + 1)

• (3x⁴ - 7)*(10x) - (12x³)*(5x² + 1)

• (3x⁴ - 7)/(10x) + (12x³)/(5x² + 1)

Multiple Choice

d/dx (sin²x) = ?

• 2sin x

• 2(sin x)(cos x)

• 2cos x

• 2x(cos x)

Multiple Choice

f(x)= 9/∛x

f'(x) =

• 9x^-1/3

• -3x^-4/3

• -9x^2/3

• -3x^2/3