Understanding Derivatives of Trigonometric Functions - Math

Card 1 of 16

Didn't Know

Knew It

1 of 1615 left

Question

$$\frac{\mathrm{d}$}{\mathrm{d} x}\sin(x)=?$

Tap to reveal answer

Answer

The derivative of a sine function does not follow the power rule. It is one that should be memorized.

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\sin(x)=\cos(x)$ .

← Didn't Know|Knew It →

Question

Find the derivative of the following function:

$f(x) = 3 ;\textup{sin}(x)$

Tap to reveal answer

Answer

The derivative of $\sin (x)$ is $\cos (x)$ . It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative).

Our function

$f(x) = 3 ;\textup{sin}(x)$

the factor of 3 does not change when we differentiate, therefore the answer is

$f(x) = 3 ;\textup{cos}(x)$

← Didn't Know|Knew It →

Question

Compute the derivative of the function $f(x) = \tan $x^{2}$$ .

Tap to reveal answer

Answer

Use the Chain Rule.

Set $u = $x^{2}$$ and substitute.

$f(x) = \tan u$

$$\frac{\mathrm{d}$ f }{\mathrm{d} u} = $\frac{\mathrm{d}$ }{\mathrm{d} u} \tan u = $\sec^{2}$ u$

$$\frac{\mathrm{d}$ u }{\mathrm{d} x} = $\frac{\mathrm{d}$ }{\mathrm{d} x} $x^{2}$ = $2x^{2-1}$ =2x$

$\frac{\mathrm{d}$ f }{\mathrm{d} x} = $\frac{\mathrm{d}$ f }{\mathrm{d} u} \cdot $\frac{\mathrm{d}$ u }{\mathrm{d} x}= $\sec^{2}$ u \cdot 2x = 2x $\sec^{2}$ $x^{2}$

← Didn't Know|Knew It →

Question

What is the second derivative of $\sin(x)$ ?

Tap to reveal answer

Answer

The derivatives of trig functions must be memorized. The first derivative is:

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\sin(x)=\cos(x)$ .

To find the second derivative, we take the derivative of our result.

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\cos(x)=-\sin(x)$ .

Therefore, the second derivative will be $-\sin(x)$ .

← Didn't Know|Knew It →

Question

$$\frac{\mathrm{d}$}{\mathrm{d} x}\sin(x)=?$

Tap to reveal answer

Answer

The derivative of a sine function does not follow the power rule. It is one that should be memorized.

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\sin(x)=\cos(x)$ .

← Didn't Know|Knew It →

Question

Find the derivative of the following function:

$f(x) = 3 ;\textup{sin}(x)$

Tap to reveal answer

Answer

The derivative of $\sin (x)$ is $\cos (x)$ . It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative).

Our function

$f(x) = 3 ;\textup{sin}(x)$

the factor of 3 does not change when we differentiate, therefore the answer is

$f(x) = 3 ;\textup{cos}(x)$

← Didn't Know|Knew It →

Question

Compute the derivative of the function $f(x) = \tan $x^{2}$$ .

Tap to reveal answer

Answer

Use the Chain Rule.

Set $u = $x^{2}$$ and substitute.

$f(x) = \tan u$

$$\frac{\mathrm{d}$ f }{\mathrm{d} u} = $\frac{\mathrm{d}$ }{\mathrm{d} u} \tan u = $\sec^{2}$ u$

$$\frac{\mathrm{d}$ u }{\mathrm{d} x} = $\frac{\mathrm{d}$ }{\mathrm{d} x} $x^{2}$ = $2x^{2-1}$ =2x$

$\frac{\mathrm{d}$ f }{\mathrm{d} x} = $\frac{\mathrm{d}$ f }{\mathrm{d} u} \cdot $\frac{\mathrm{d}$ u }{\mathrm{d} x}= $\sec^{2}$ u \cdot 2x = 2x $\sec^{2}$ $x^{2}$

← Didn't Know|Knew It →

Question

What is the second derivative of $\sin(x)$ ?

Tap to reveal answer

Answer

The derivatives of trig functions must be memorized. The first derivative is:

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\sin(x)=\cos(x)$ .

To find the second derivative, we take the derivative of our result.

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\cos(x)=-\sin(x)$ .

Therefore, the second derivative will be $-\sin(x)$ .

← Didn't Know|Knew It →

Question

Find the derivative of the following function:

$f(x) = 3 ;\textup{sin}(x)$

Tap to reveal answer

Answer

The derivative of $\sin (x)$ is $\cos (x)$ . It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative).

Our function

$f(x) = 3 ;\textup{sin}(x)$

the factor of 3 does not change when we differentiate, therefore the answer is

$f(x) = 3 ;\textup{cos}(x)$

← Didn't Know|Knew It →

Question

Compute the derivative of the function $f(x) = \tan $x^{2}$$ .

Tap to reveal answer

Answer

Use the Chain Rule.

Set $u = $x^{2}$$ and substitute.

$f(x) = \tan u$

$$\frac{\mathrm{d}$ f }{\mathrm{d} u} = $\frac{\mathrm{d}$ }{\mathrm{d} u} \tan u = $\sec^{2}$ u$

$$\frac{\mathrm{d}$ u }{\mathrm{d} x} = $\frac{\mathrm{d}$ }{\mathrm{d} x} $x^{2}$ = $2x^{2-1}$ =2x$

$\frac{\mathrm{d}$ f }{\mathrm{d} x} = $\frac{\mathrm{d}$ f }{\mathrm{d} u} \cdot $\frac{\mathrm{d}$ u }{\mathrm{d} x}= $\sec^{2}$ u \cdot 2x = 2x $\sec^{2}$ $x^{2}$

← Didn't Know|Knew It →

Question

$$\frac{\mathrm{d}$}{\mathrm{d} x}\sin(x)=?$

Tap to reveal answer

Answer

The derivative of a sine function does not follow the power rule. It is one that should be memorized.

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\sin(x)=\cos(x)$ .

← Didn't Know|Knew It →

Question

What is the second derivative of $\sin(x)$ ?

Tap to reveal answer

Answer

The derivatives of trig functions must be memorized. The first derivative is:

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\sin(x)=\cos(x)$ .

To find the second derivative, we take the derivative of our result.

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\cos(x)=-\sin(x)$ .

Therefore, the second derivative will be $-\sin(x)$ .

← Didn't Know|Knew It →

Question

Find the derivative of the following function:

$f(x) = 3 ;\textup{sin}(x)$

Tap to reveal answer

Answer

The derivative of $\sin (x)$ is $\cos (x)$ . It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative).

Our function

$f(x) = 3 ;\textup{sin}(x)$

the factor of 3 does not change when we differentiate, therefore the answer is

$f(x) = 3 ;\textup{cos}(x)$

← Didn't Know|Knew It →

Question

Compute the derivative of the function $f(x) = \tan $x^{2}$$ .

Tap to reveal answer

Answer

Use the Chain Rule.

Set $u = $x^{2}$$ and substitute.

$f(x) = \tan u$

$$\frac{\mathrm{d}$ f }{\mathrm{d} u} = $\frac{\mathrm{d}$ }{\mathrm{d} u} \tan u = $\sec^{2}$ u$

$$\frac{\mathrm{d}$ u }{\mathrm{d} x} = $\frac{\mathrm{d}$ }{\mathrm{d} x} $x^{2}$ = $2x^{2-1}$ =2x$

$\frac{\mathrm{d}$ f }{\mathrm{d} x} = $\frac{\mathrm{d}$ f }{\mathrm{d} u} \cdot $\frac{\mathrm{d}$ u }{\mathrm{d} x}= $\sec^{2}$ u \cdot 2x = 2x $\sec^{2}$ $x^{2}$

← Didn't Know|Knew It →

Question

$$\frac{\mathrm{d}$}{\mathrm{d} x}\sin(x)=?$

Tap to reveal answer

Answer

The derivative of a sine function does not follow the power rule. It is one that should be memorized.

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\sin(x)=\cos(x)$ .

← Didn't Know|Knew It →

Question

What is the second derivative of $\sin(x)$ ?

Tap to reveal answer

Answer

The derivatives of trig functions must be memorized. The first derivative is:

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\sin(x)=\cos(x)$ .

To find the second derivative, we take the derivative of our result.

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\cos(x)=-\sin(x)$ .

Therefore, the second derivative will be $-\sin(x)$ .

← Didn't Know|Knew It →

Knew It

Understanding Derivatives of Trigonometric Functions - Math

The derivative of a sine function does not follow the power rule. It is one that should be memorized. .

Find the derivative of the following function:

The derivative of is. It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative). Our function the factor of 3 does not change when we differentiate, therefore the answer is

Compute the derivative of the function .

Use the Chain Rule. Set and substitute.

What is the second derivative of ?

The derivatives of trig functions must be memorized. The first derivative is: . To find the second derivative, we take the derivative of our result. . Therefore, the second derivative will be .

The derivative of a sine function does not follow the power rule. It is one that should be memorized. .

Find the derivative of the following function:

The derivative of is. It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative). Our function the factor of 3 does not change when we differentiate, therefore the answer is

Compute the derivative of the function .

Use the Chain Rule. Set and substitute.

What is the second derivative of ?

The derivatives of trig functions must be memorized. The first derivative is: . To find the second derivative, we take the derivative of our result. . Therefore, the second derivative will be .

Find the derivative of the following function:

The derivative of is. It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative). Our function the factor of 3 does not change when we differentiate, therefore the answer is

Compute the derivative of the function .

Use the Chain Rule. Set and substitute.

The derivative of a sine function does not follow the power rule. It is one that should be memorized. .

What is the second derivative of ?

The derivatives of trig functions must be memorized. The first derivative is: . To find the second derivative, we take the derivative of our result. . Therefore, the second derivative will be .

Find the derivative of the following function:

The derivative of is. It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative). Our function the factor of 3 does not change when we differentiate, therefore the answer is

Compute the derivative of the function .

Use the Chain Rule. Set and substitute.

The derivative of a sine function does not follow the power rule. It is one that should be memorized. .

What is the second derivative of ?

The derivatives of trig functions must be memorized. The first derivative is: . To find the second derivative, we take the derivative of our result. . Therefore, the second derivative will be .

The derivative of a sine function does not follow the power rule. It is one that should be memorized.

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\sin(x)=\cos(x)$ .

Find the derivative of the following function:

$f(x) = 3 ;\textup{sin}(x)$

Compute the derivative of the function $f(x) = \tan $x^{2}$$ .

What is the second derivative of $\sin(x)$ ?

The derivative of a sine function does not follow the power rule. It is one that should be memorized.

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\sin(x)=\cos(x)$ .

Find the derivative of the following function:

$f(x) = 3 ;\textup{sin}(x)$

Compute the derivative of the function $f(x) = \tan $x^{2}$$ .

What is the second derivative of $\sin(x)$ ?

Find the derivative of the following function:

$f(x) = 3 ;\textup{sin}(x)$

Compute the derivative of the function $f(x) = \tan $x^{2}$$ .

The derivative of a sine function does not follow the power rule. It is one that should be memorized.

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\sin(x)=\cos(x)$ .

What is the second derivative of $\sin(x)$ ?

Find the derivative of the following function:

$f(x) = 3 ;\textup{sin}(x)$

Compute the derivative of the function $f(x) = \tan $x^{2}$$ .

The derivative of a sine function does not follow the power rule. It is one that should be memorized.

$$\frac{\mathrm{d}$ }{\mathrm{d} x}\sin(x)=\cos(x)$ .

What is the second derivative of $\sin(x)$ ?