Question

Log cos ^(-1) (x) differentiate this function

Answer 1

Answer:

When tackling the derivative of inverse trig functions. I prefer to rearrange and use Implicit differentiation as I always get the inverse derivatives muddled up, and this way I do not need to remember the inverse derivatives. If you can remember the inverse derivatives then you can use the chain rule.

Let

y

=

cos

−

1

(

x

)

⇔

cos

y

=

x

Differentiate Implicitly:

−

sin

y

d

y

d

x

=

1

..... [1]

Using the

sin

/

cos

identity;

sin

2

y

+

cos

2

y

≡

1

∴

sin

2

y

+

x

2

=

1

∴

sin

2

y

=

1

−

x

2

∴

sin

y

=

√

1

−

x

2

Substituting into [1]

∴

−

√

1

−

x

2

d

y

d

x

=

1

∴

d

y

d

x

=

−

1

√

1

−

x

2

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