d/dx tan(x)
To find the derivative of the function f(x) = tan(x), we’ll use the quotient rule
To find the derivative of the function f(x) = tan(x), we’ll use the quotient rule.
The quotient rule states that if we have a function in the form f(x) = g(x) / h(x), then its derivative is given by:
f'(x) = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]^2
In this case, g(x) is the numerator and h(x) is the denominator. We have g(x) = 1 and h(x) = cos(x).
Now, let’s find the derivatives of g(x) and h(x):
g'(x) = 0 (the derivative of a constant is 0)
h'(x) = -sin(x) (the derivative of cos(x) is -sin(x) by the chain rule)
Substituting these into the quotient rule formula, we get:
f'(x) = [0 * cos(x) – 1 * (-sin(x))] / [cos(x)]^2
= [sin(x)] / [cos(x)]^2
Alternatively, we can simplify this further by using the identity sin(x) = cos(x) * tan(x):
f'(x) = [cos(x) * tan(x)] / [cos(x)]^2
= tan(x) / cos(x)
Since cos(x) is the reciprocal of sec(x), we can rewrite the formula as:
f'(x) = tan(x) / cos(x) = tan(x) * sec(x)
Therefore, the derivative of tan(x) with respect to x is tan(x) * sec(x).
More Answers:
Mastering u-substitution to solve the integral of sec(u) duThe Chain Rule: Finding the Derivative of sin(x) with Respect to x and the Surprising Result
Mastering the Chain Rule: Simplifying the Derivative of cos(x)