How do you differentiate #f(x) = sec(x^2 + 1)^2#?

2 Answers
Trevor Ryan. · Becca M.
Aug 24, 2015

#4x (x^2 + 1) * sec(x^2 + 1)^2 * tan(X^2 + 1)^2#

Explanation:

The derivative of the sec of a function is the sec of the function multiplied by the tan of the function multiplied by the derivative of the function. And the derivative of a function to a power is, by the power rule, the power times the function raised to one less than the given power, times the derivative of the function. And the derivative of a polynomial function is the power times the coefficient times the variable raised to one less than the given power.

Jim H
Sep 1, 2015

Here is the "How" using notation.

Explanation:

#f(x) = sec(x^2+1)^2#

#d/dx(secu) = secutanu (du)/dx#, so we get

#f'(x) = sec(x^2+1)^2tan(x^2+1)^2 * d/dx((x^2+1)^2)#

# = sec(x^2+1)^2tan(x^2+1)^2 * 2(x^2+1)d/dx(x^2+1)#

# = sec(x^2+1)^2tan(x^2+1)^2 * 2(x^2+1)(2x)#

# = 4x(x^2+1) sec(x^2+1)^2tan(x^2+1)^2 #

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