Find the derivative of the function.h(t)=sin(t)cos(t)h(t)=\sin\left(t\right)\cos\left(t\right)

−sin⁡2(t)+cos⁡2(t)-\$$sin^2$$\left(t\right)+\$$cos^2$$\left(t\right)−sin2(t)+cos2(t)

Find the derivative of the function. h ( t ) = sin ( t ) cos ( t ) h(t)=\sin\left(t\right)\cos\left(t\right)

Here, we're asked to find the derivative of the function h of t equals sine t times cosine t. Now here, I have 2 trig functions multiplying each other, which means that to find the derivative, I need to use the product rule here. So let's go ahead and jump in. Now in order to find my derivative here, we're just taking the first derivative, h prime of t of sine t times cosine t. So using my product rule here, I'm going to take that left function sine of t and then multiply it by the derivative of that right function. Now the derivative of cosine is a negative sine, so I'm gonna be multiplying it by that here. Now that's my first term. Using the product rule, I'm going to add that together with my second term, which is going to be that right function, which is that cosine. So I have cosine of t times the derivative of that left hand function, sine of t. Remember that when using the product rule, we're always going to be taking one original function and then multiplying it by the derivative of the other function. Now here, the derivative of the sine of t is just the positive cosine of t. So this is what I have here for the product rule. But we can do a bit of simplification here because I have sine multiplying negative sine, which means that this is a negative sine squared of t. And then that's multiplying cosine times cosine, which is positive cosine squared of t. Simplifying this as much as possible for my derivative here, I end up getting negative sine squared of t plus cosine squared of t. Now, you could choose to rewrite this as cosine squared t minus sine squared t if you don't like having that negative sign in front. But either of these two answers would be correct. Now, you may be tempted here to simplify this using some trig identities. But because this is a negative sign squared, you can't do that. If it was a positive, then you could, but we don't have to worry about that here. So this is our final answer. Now as you continue to take more and more derivatives dealing with trig functions, you're going to have to refer to all of those derivatives less and less as you commit them to memory. So let's continue getting some more practice dealing with derivatives of trick functions. Thanks for watching, and I'll see you in the next one.

Find the derivative of the function.h(t)=sin⁡(t)cos⁡(t)h(t)=\sin\left(t\right)\cos\left(t\right)

−sin⁡2(t)+cos⁡2(t)-\$$sin^2$$\left(t\right)+\$$cos^2$$\left(t\right)−sin2(t)+cos2(t)

2sin⁡(t)cos⁡(t)2\sin\left(t\right)\cos\left(t\right)2sin(t)cos(t)

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0. Functions7h 55m

Introduction to Functions
18m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
36m

Exponential & Logarithmic Equations
35m

Introduction to Trigonometric Functions
38m

Graphs of Trigonometric Functions
44m

Trigonometric Identities
47m

Inverse Trigonometric Functions
48m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation3h 18m

Basic Rules of Differentiation
39m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

Derivatives of Trig Functions
40m

The Chain Rule
49m

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Motion Analysis
36m

Implicit Differentiation
27m

Related Rates
59m

Linearization
13m

Differentials
21m

5. Graphical Applications of Derivatives6h 2m

Intro to Extrema
23m

Finding Global Extrema
50m

The First Derivative Test
1h 5m

Concavity
1h 1m

The Second Derivative Test
31m

Curve Sketching
44m

Applied Optimization
1h 25m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m

Derivatives of Exponential & Logarithmic Functions
1h 52m

Logarithmic Differentiation
20m

Derivatives of Inverse Trigonometric Functions
24m

7. Antiderivatives & Indefinite Integrals1h 26m

Antiderivatives
17m

Indefinite Integrals
25m

Integrals of Trig Functions
15m

Initial Value Problems
27m

8. Definite Integrals4h 44m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 21m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
37m

Average Value of a Function
21m

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1h 19m

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1h 18m

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1h 8m

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1h 13m

Work
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Integrals of Exponential Functions
40m

Integrals Involving Logarithmic Functions
58m

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52m

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2h 32m

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39m

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Calculus in Polar Coordinates
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3. Techniques of Differentiation

Derivatives of Trig Functions

Calculus

3. Techniques of Differentiation

Derivatives of Trig Functions

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Find the derivative of the function.
$h (t) = \sin (t) \cos (t)$

$0$

$1$

$2$\sin\[\left$(t$\right$)$\cos\]\left$(t$\right$)$

$-$\sin$^2$\left$(t$\right$)+$\cos$^2$\left$(t$\right$)$

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Identify the function h(t) = sin(t)cos(t). Notice that this is a product of two functions: u(t) = sin(t) and v(t) = cos(t).

To find the derivative of h(t), apply the product rule for differentiation, which states that if h(t) = u(t)v(t), then h'(t) = u'(t)v(t) + u(t)v'(t).

Differentiate u(t) = sin(t) to get u'(t) = cos(t).

Differentiate v(t) = cos(t) to get v'(t) = -sin(t).

Substitute u(t), v(t), u'(t), and v'(t) into the product rule formula: h'(t) = cos(t)cos(t) + sin(t)(-sin(t)). Simplify the expression to get the derivative.