Learn How To Find Derivative Of Sin(X) Using The Chain Rule

d/dx(tanx)

sec²x

To find the derivative of tanx with respect to x, we use the quotient rule:

Let f(x) = tanx

Then, f'(x) = [(secx)^2 * d/dx(x)] – [tanx * d/dx(secx)]

Using the product rule, we find that d/dx(secx) = secx * tanx

Substituting this back into the original equation, we have:

f'(x) = [(secx)^2 * 1] – [tanx * (secx * tanx)]

Simplifying the second term, we get:

f'(x) = (secx)^2 – (tanx)^2(secx)^2

Recall that (tanx)^2 + 1 = (secx)^2

Substituting this into the second term, we have:

f'(x) = (secx)^2 – (1/(cosx)^2)(cosx)^2

Simplifying further, we get:

f'(x) = (secx)^2 – 1

Recall that secx = 1/cosx

Substituting this into the equation, we have:

f'(x) = (1/cosx)^2 – 1

Simplifying again, we get:

f'(x) = (1 – cos^2x)/cos^2x

Recall that sin^2x + cos^2x = 1

Substituting this into the equation, we have:

f'(x) = (sin^2x)/cos^2x

Recall that tanx = sinx/cosx

Substituting this into the equation, we have:

f'(x) = tan^2x

Therefore, the derivative of tanx with respect to x is tan^2x.

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