d/dx[tanx]
To find the derivative of tan(x), we’ll use the quotient rule
To find the derivative of tan(x), we’ll use the quotient rule. The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative of f(x) with respect to x is given by:
f'(x) = (g'(x)h(x) – g(x)h'(x)) / (h(x))^2
In this case, g(x) = sin(x) and h(x) = cos(x). So, we’ll need to find the derivatives of sin(x) and cos(x) and plug them into the quotient rule.
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).
Using the quotient rule, we can find the derivative of tan(x):
tan'(x) = (cos(x)(cos(x)) – sin(x)(-sin(x))) / (cos(x))^2
Simplifying further:
tan'(x) = (cos^2(x) + sin^2(x)) / (cos^2(x))
Remember that sin^2(x) + cos^2(x) = 1. So, the formula becomes:
tan'(x) = 1 / (cos^2(x))
Alternatively, we can write this as:
tan'(x) = sec^2(x)
Therefore, the derivative of tan(x) is sec^2(x).
I hope this explanation helps. Let me know if you have any further questions!
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