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How do you differentiate #y=x^2(sin^-1x)^3#?

Answer 1

Samantha Adkison

Below

#y=x^2((sinx)^(-1))^3#

#(dy)/(dx)=x^2*3((sinx)^(-1))^2*1/sqrt(1-x^2)+2x*((sinx)^(-1))^3#

#(dy)/(dx)=(3x^2((sinx)^(-1))^2)/sqrt(1-x^2)+2x((sinx)^(-1))^3#

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Answer 2

Paisley Johnson

To differentiate ( y = x^2(\sin^{-1}x)^3 ), you can use the product rule and chain rule. Here's the process:

Let ( u = x^2 ) and ( v = (\sin^{-1}x)^3 ).
Find the derivatives of ( u ) and ( v ) with respect to ( x ): ( \frac{du}{dx} = 2x ) and ( \frac{dv}{dx} = 3(\sin^{-1}x)^2 \cdot \frac{1}{\sqrt{1 - x^2}} ).
Apply the product rule: ( \frac{d}{dx}(uv) = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx} ).
Substitute ( u ), ( v ), ( \frac{du}{dx} ), and ( \frac{dv}{dx} ) into the product rule formula.
Simplify the expression.

The result after simplification will be the derivative of ( y ) with respect to ( x ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.