d/dx tanx
To find the derivative of the function f(x) = tan(x), we can use the definition of the derivative and some trigonometric identities
To find the derivative of the function f(x) = tan(x), we can use the definition of the derivative and some trigonometric identities.
The derivative of a function represents its rate of change with respect to the variable x. In this case, we are finding the derivative of the tangent function, which gives us the rate of change of tan(x) as x varies.
Using the definition of the derivative, we have:
f'(x) = lim(h → 0) [f(x + h) – f(x)] / h
Let’s proceed with finding the derivative step by step:
Step 1: Express tan(x) in terms of sin(x) and cos(x).
The tangent function can be expressed as the ratio of the sine and cosine functions:
tan(x) = sin(x) / cos(x)
Step 2: Rewrite the expression to simplify it.
f(x + h) = tan(x + h) = sin(x + h) / cos(x + h)
f(x) = tan(x) = sin(x) / cos(x)
Step 3: Calculate the difference quotient.
[f(x + h) – f(x)] / h = [sin(x + h) / cos(x + h) – sin(x) / cos(x)] / h
Now, let’s simplify this expression further.
Step 4: Combine the fractions with a common denominator.
= [sin(x + h)cos(x) – sin(x)cos(x + h)] / [cos(x + h)cos(x)] / h
Step 5: Expand the terms.
= [sin(x)cos(h) + cos(x)sin(h) – sin(x)cos(x) – sin(x)cos(h)] / [cos(x)cos(h)] / h
= [cos(x)sin(h) – sin(x)cos(x)] / [cos(x)cos(h)] / h
Step 6: Divide through by h.
= [cos(x)sin(h) – sin(x)cos(x)] / [cos(x)cos(h)]
Step 7: Take the limit as h approaches 0.
As h approaches 0, sin(h) and cos(h) approach 0. Therefore, we can substitute sin(h) = 0 and cos(h) = 1.
= [cos(x)(0) – sin(x)cos(x)] / [cos(x)(1)]
= -sin(x) / cos(x)
Step 8: Simplify the expression further using a trigonometric identity.
Recall that the tangent function is equal to the ratio of sine and cosine:
= -sin(x) / cos(x) = -tan(x)
Therefore, we have found that the derivative of tan(x) is equal to -tan(x).
In differential notation, we can write it as:
d/dx tan(x) = -tan(x)
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