Differentiating Trigonometric Functions: Find the Derivatives

In summary: First, take the derivative of sin5x, which is 5cos5x. Second, multiply it by (sin x)5 to get 5cos5x×(sin x)5.For #4, use the chain rule.y = √(4sin^2x+9cos^2x)u = 4sin^2x+9cos^2xy = u1/2y' = (1/2)u^(1/2-1)×u'= (1/2)(4sin^2x+9cos^2x)^(-1/2)×(8sinxcosx-18sinxcosx)= (
  • #1
nephi37
9
0
find the derivatives

of differentiation of trigonometric functions

1. y=cos(3x^2+8x-2)

2. y=tan^3 2x

3. y=sin5x sin^5 x

4. y=Square root of 4sin^2x+9cos^2x

help here please..

i can't understand trigonometric functions

sorry admin or moderator, i just search the net on how to do this problems.. and this are my answer

for
1. -6x+8sin(3x^2+8x-2)

2. 6 tan 2x sec^2 2x

3. 5 cos 5x 5 sin x cos x or 5 cos 5x 5 sin^4 x cos x

4. 8 sec^4 2x tan 2x

Are my answers correct.. Help me please
 
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  • #2
let us do it one by one:
(1) is not correct.
 
  • #3
(2) is basically correct except that it should be tan^2 since the derivative of tan^3 is 3tan^2...
 
  • #4
nephi37 said:
1. y=cos(3x^2+8x-2)

re 1.

1st step: derivative of cos(...)
2nd step: MULTIPLY by the derivative of (3x^2+8x-2)
 
  • #5
Actually, (1) is nearly correct.

Use parentheses around (6x+8), the derivative of 3x^2+8x-2 .

Start (3) by using the product rule.


By the Way: Until you get the hang of this, it's probably best to write trig functions raised to a power as:

(sin(x))5 rather than sin5(x),

(tan(2x))3 rather than tan3(2x)
 
Last edited:
  • #6
so my answer in number 1 should be

-(6x+8)sin(3x^2+8x-2)

and in number to should be

6 tan^2 2x sec^2 2x

Am i correct.. or still wrong :)
 
  • #7
in number 1

-sin(3x^2+8x-2)(6x+8) then multiply -sin to (6x+8)

-(6x+8)sin(3x^2+8x-2)

am i correct now guys :)
 
  • #8
nephi37 said:
in number 1

-sin(3x^2+8x-2)(6x+8) then multiply -sin to (6x+8)

-(6x+8)sin(3x^2+8x-2)

am i correct now guys :)
Either one is OK !
 
  • #9
thanks sammy..

its now 9 am here we nid to pass this by 10am

thanks for help...
but how about my number 3 and number 4

is that correct? I am not sure what I've done..

especially number 4 with square root thing :)
 
  • #10
Look at √x as x1/2,

So y=√(4sin^2x+9cos^2x)

becomes y=(4sin^2x+9cos^2x)1/2.

Have fun.
 
  • #11
For #3. Use the product rule.

y = {sin5x}×{(sin x)5}
 

What is the definition of a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is calculated as the slope of the tangent line at that point.

What are the basic rules for finding the derivatives of trigonometric functions?

The basic rules for finding the derivatives of trigonometric functions are the power rule, product rule, quotient rule, and chain rule. The power rule states that the derivative of a function raised to a power is equal to the power multiplied by the derivative of the function. The product rule and quotient rule are used to find the derivatives of products and quotients of functions, respectively. The chain rule is used for finding the derivatives of composite functions.

How do I differentiate trigonometric functions?

To differentiate trigonometric functions, you can use the basic rules mentioned above, as well as the specific rules for each trigonometric function. For example, the derivative of sine is cosine, the derivative of cosine is negative sine, and the derivative of tangent is secant squared. You may also need to use trigonometric identities to simplify the function before differentiating.

What is the difference between implicit and explicit differentiation?

Implicit differentiation is used when the dependent variable is not explicitly defined in terms of the independent variable. In this case, the derivative is found by treating the dependent variable as a function of the independent variable. On the other hand, explicit differentiation is used when the dependent variable is explicitly defined in terms of the independent variable. In this case, the derivative is found using the basic rules of differentiation.

How can I check if my derivative is correct?

To check if your derivative is correct, you can use various methods such as plugging in values for the independent variable and comparing the results to the original function, graphing the original function and its derivative to see if they have the same slope at different points, or using online tools or software to calculate the derivative and compare it to your result.

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