Question
Differentiate the function \( G(x)=-7 \sqrt{x} \sec x \)
Ask by Ortiz Boone. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The derivative of \( G(x) = -7 \sqrt{x} \sec x \) is \( G'(x) = -\frac{7 \sec x}{2\sqrt{x}} - 7\sqrt{x} \sec x \tan x \).
Solution
We are given G(x) = -7 √x sec x. To differentiate G(x), we will use the product rule. Recall that the product rule states:
(uv)' = u'v + uv'
Let u(x) = -7 √x and v(x) = sec x.
Step 1: Differentiate u(x).
u(x) = -7 x^(1/2)
u'(x) = -7 * (1/2) x^(-1/2) = -7/(2√x)
Step 2: Differentiate v(x).
v(x) = sec x
v'(x) = sec x tan x
Step 3: Apply the product rule.
G'(x) = u'(x)v(x) + u(x)v'(x)
G'(x) = (-7/(2√x)) sec x + (-7√x)(sec x tan x)
Step 4: Factor and simplify if desired.
G'(x) = -7 sec x/(2√x) - 7√x (sec x tan x)
Thus, the derivative of G(x) is:
G'(x) = -7 sec x/(2√x) - 7√x sec x tan x.
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Simplify this solution Extra Insights
To differentiate the function \( G(x) = -7 \sqrt{x} \sec x \), we can use the product rule since it is the product of two functions: \( -7 \sqrt{x} \) and \( \sec x \). First, let's find the derivatives of each part: 1. The derivative of \( -7 \sqrt{x} \) is \( -\frac{7}{2\sqrt{x}} \). 2. The derivative of \( \sec x \) is \( \sec x \tan x \). Now, applying the product rule \( (uv)' = u'v + uv' \): \[ G'(x) = \left(-\frac{7}{2\sqrt{x}}\right) \sec x + (-7\sqrt{x})(\sec x \tan x) \] Putting it all together, we get: \[ G'(x) = -\frac{7}{2\sqrt{x}} \sec x - 7\sqrt{x} \sec x \tan x \] That’s the derivative of \( G(x) \)!
