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Question:
Differentiate sec^2(x)
Applying the Chain Rule for Derivatives:
Sometimes, in the study of calculus, we are given a composite function, or a function of the form f(g(x)), and asked to differentiate it. If we can identify f(x) and g(x) in these instances, we have a formula called the chain rule for derivatives that we can use to find the derivative in a straightforward manner.
Useful Formulas and Vocabulary
Derivative of a Function: The derivative of a function f(x) is denoted as f ‘ (x), and f ‘ (a) is equivalent to the slope of the tangent line to f(x) at x = a. This can also be described as the instantaneous rate of change of f(x) at x = a.
The Chain Rule for Derivatives: The chain rule for derivatives is a rule that allows us to find the derivative of a composition of functions, where a composition of functions is a function of the form
f
(
g
(
x
)
)
.
Formulas for Derivatives of Trigonometric Functions: To compute derivatives of trigonometric functions, we have well-known formulas that we can use. Each trigonometric function has its own specific derivative formula.
Derivative of
x
n
: The derivative of
x
n
is given by the formula
n
x
n
−
1
.
Answer and Explanation:
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The derivative of sec2 (x) is 2sec2 (x) tan (x).
The chain rule states that the derivative of f(g(x)) is equal to f ‘ (g(x)) ⋅ g ‘ (x)….
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