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Find the derivative of $\sec\left(\frac{4}{x}\right)^2$

Step-by-step Solution

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Final Answer

$\frac{-8\sec\left(\frac{4}{x}\right)^2\tan\left(\frac{4}{x}\right)}{x^2}$
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Step-by-step Solution

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The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$2\sec\left(\frac{4}{x}\right)\frac{d}{dx}\left(\sec\left(\frac{4}{x}\right)\right)$

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$2\sec\left(\frac{4}{x}\right)\frac{d}{dx}\left(\sec\left(\frac{4}{x}\right)\right)$

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Learn how to solve differential calculus problems step by step online. Find the derivative of sec(4/x)^2. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}. Taking the derivative of secant function: \frac{d}{dx}\left(\sec(x)\right)=\sec(x)\cdot\tan(x)\cdot D_x(x). When multiplying two powers that have the same base (\sec\left(\frac{4}{x}\right)), you can add the exponents. Apply the quotient rule for differentiation, which states that if f(x) and g(x) are functions and h(x) is the function defined by {\displaystyle h(x) = \frac{f(x)}{g(x)}}, where {g(x) \neq 0}, then {\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}.

Final Answer

$\frac{-8\sec\left(\frac{4}{x}\right)^2\tan\left(\frac{4}{x}\right)}{x^2}$

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Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind derivative of sec(4/x)^2 using the product ruleFind derivative of sec(4/x)^2 using the quotient ruleFind derivative of sec(4/x)^2 using logarithmic differentiationFind derivative of sec(4/x)^2 using the definition

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Plotting: $\frac{-8\sec\left(\frac{4}{x}\right)^2\tan\left(\frac{4}{x}\right)}{x^2}$

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7
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0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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Main Topic: Differential Calculus

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.

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