𝑑/𝑑𝑥[tan 𝑥]
To find the derivative of tan(x), we can use the quotient rule
To find the derivative of tan(x), we can use the quotient rule.
Let’s start by writing tan(x) as sin(x)/cos(x):
tan(x) = sin(x) / cos(x)
Now, let’s apply the quotient rule:
d/dx [sin(x)/cos(x)] = (cos(x) * d/dx[sin(x)] – sin(x) * d/dx[cos(x)]) / (cos(x))^2
The derivative of sin(x) with respect to x is cos(x), and the derivative of cos(x) with respect to x is -sin(x):
d/dx [sin(x)] = cos(x)
d/dx [cos(x)] = -sin(x)
Plugging these values into the quotient rule formula:
d/dx [tan(x)] = (cos(x) * cos(x) – sin(x) * -sin(x)) / (cos(x))^2
= (cos^2(x) + sin^2(x)) / (cos^2(x))
= 1 / cos^2(x)
Since cos^2(x) is equal to 1/(sec^2(x)), we can simplify the expression:
d/dx [tan(x)] = 1 / cos^2(x) = 1 / (1/sec^2(x)) = sec^2(x)
So, the derivative of tan(x) with respect to x is sec^2(x).
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