Find the derivatives of the inverse functions of the following y=xcosx
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Answers
Answer:
ANSWER
y=x
2
⋅e
x
Differentiating w.r.t. x, we get
dx
dy
=
dx
d
(x
2
⋅e
x
)
=x
2
dx
d
(e
x
)+e
x
dx
d
(x
2
)
=x
2
⋅e
x
+e
x
×2x
=xe
x
(x+2)
The derivative of inverse function of y=f(x) is given by
dy
dx
=
(
dx
dy
)
1
=
xe
x
(x+2)
1
An
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Each of the six basic trigonometric functions have corresponding inverse functions when appropriate restrictions are placed on the domain of the original functions. All the inverse trigonometric functions have derivatives, which are summarized as follows: Example 1: Find f′( x) if f( x) = cos −1(5 x).
Use the inverse function theorem to find the derivative of g(x)=3√x.
The function g(x)=3√x is the inverse of the function f(x)=x3. Since g′(x)=1f′(g(x)), begin by finding f′(x). Thus,
f′(x)=3x3.
f′(g(x))=3(3√x)2=3x2/3.
g′(x)=13x2/3=13x−2/3.