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# Find the derivative of $\sin\left(2x\right)$ using the definition

## Step-by-step Solution

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###  Videos

$2\cos\left(2x\right)$
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## Step-by-step Solution

Problem to solve:

$\frac{d}{dx}\left(sin\left(2x\right)\right)$

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1

Find the derivative of $\sin\left(2x\right)$ using the definition. Apply the definition of the derivative: $\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. The function $f(x)$ is the function we want to differentiate, which is $\sin\left(2x\right)$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get

$\lim_{h\to0}\left(\frac{\sin\left(2\left(x+h\right)\right)-\sin\left(2x\right)}{h}\right)$
2

Multiply the single term $2$ by each term of the polynomial $\left(x+h\right)$

$\lim_{h\to0}\left(\frac{\sin\left(2x+2h\right)-\sin\left(2x\right)}{h}\right)$

Learn how to solve definition of derivative problems step by step online.

$\lim_{h\to0}\left(\frac{\sin\left(2\left(x+h\right)\right)-\sin\left(2x\right)}{h}\right)$

Learn how to solve definition of derivative problems step by step online. Find the derivative of sin(2x) using the definition. Find the derivative of \sin\left(2x\right) using the definition. Apply the definition of the derivative: \displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}. The function f(x) is the function we want to differentiate, which is \sin\left(2x\right). Substituting f(x+h) and f(x) on the limit, we get. Multiply the single term 2 by each term of the polynomial \left(x+h\right). Using the sine of a sum formula: \sin(\alpha\pm\beta)=\sin(\alpha)\cos(\beta)\pm\cos(\alpha)\sin(\beta), where angle \alpha equals 2x, and angle \beta equals 2h. Factoring by \sin\left(2x\right).

$2\cos\left(2x\right)$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind derivdef(sin(2x)) using the product ruleFind derivdef(sin(2x)) using the quotient ruleFind derivdef(sin(2x)) using logarithmic differentiation
SnapXam A2

### beta Got a different answer? Verify it!

Go!
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0
a
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x
y
z
.
(◻)
+
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×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

$\frac{d}{dx}\left(sin\left(2x\right)\right)$