In each case, find f'(x).
Problem 1 :
Solution :
Find the derivative with respect to x using quotient rule.
Problem 2 :
f(x) = 2sin^{-1}(4x^{3})
Solution :
f(x) = 2sin^{-1}(4x^{3})
Let u = 4x^{3}.
Now, we have
f(x) = 2sin^{-1}u and u = 4x^{3}
Find the derivative with respect to x using the method derivative of a functiuon of function.
Problem 3 :
Solution :
Now, we have
Find the derivative with respect to x using the method derivative of a functiuon of function.
Problem 4 :
f(x) = ln 4x^{2}
Solution :
f(x) = ln 4x^{2}
Use the properties of logarithms on the right side.
f(x) = ln 4 + ln x^{2}
f(x) = ln 4 + 2ln x
Find the derivative with respect to x.
Problem 5 :
Solution :
f(x) = 7csc x
Find the derivative with respect to x.
f'(x) = 7(-csc x ⋅ cot x)
Problem 6 :
f(x) = 5sec^{2}x
Solution :
Let u = sec x.
Now, we have
f(x) = u^{2} and u = sec x
Find the derivative with respect to x using the method derivative of a functiuon of function.
f'(x) = 2u ⋅ (sec x ⋅ tan x)
f'(x) = 2sec x ⋅ (sec x ⋅ tan x)
f'(x) = 2sec^{2}x ⋅ tan x
Problem 7 :
f(x) = 4^{x}
We know that
(a^{x})' = a^{x} ⋅ ln a
where a is a constant.
Solution :
f(x) = 4^{x} ⋅ ln 4
Problem 8 :
f(x) = ln (5x - 4)
Solution :
Let u = 5x - 4.
Now, we have
f(x) = ln u and u = 5x - 4
Find the derivative with respect to x using the method derivative of a functiuon of function.
Problem 9 :
Solution :
Problem 10 :
Solution :
Problem 11 :
Solution :
Problem 12 :
Solution :
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