Find the derivative of #y=3tan^-1(x+sqrt(1+x^2))#?
We know that.
Here,
Substitute, #x=tantheta=>theta=tan^-1x,where,thetain (- pi/2,pi/2)#
Now,
So,
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[ \frac{d}{dx} \left[ \tan^{-The derivative of ( y = 3\tan^{-1}(x+\sqrt{1+x^2}) ) is ( \frac{3}{1 + (x+\To find the derivative of ( y = 3 \tan^{-1}(x + \sqrt{1 + x^2}) ), we'll use the chain rule:
[ \frac{d}{dx} \left[ \tan^{-1}(The derivative of ( y = 3\tan^{-1}(x+\sqrt{1+x^2}) ) is ( \frac{3}{1 + (x+\sqrt{To find the derivative of ( y = 3 \tan^{-1}(x + \sqrt{1 + x^2}) ), we'll use the chain rule:
[ \frac{d}{dx} \left[ \tan^{-1}(uThe derivative of ( y = 3\tan^{-1}(x+\sqrt{1+x^2}) ) is ( \frac{3}{1 + (x+\sqrt{1To find the derivative of ( y = 3 \tan^{-1}(x + \sqrt{1 + x^2}) ), we'll use the chain rule:
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[ \frac{d}{dx} \left[ \tan^{-1}(u) \right]The derivative of ( y = 3\tan^{-1}(x+\sqrt{1+x^2}) ) is ( \frac{3}{1 + (x+\sqrt{1+x^2To find the derivative of ( y = 3 \tan^{-1}(x + \sqrt{1 + x^2}) ), we'll use the chain rule:
[ \frac{d}{dx} \left[ \tan^{-1}(u) \right] =The derivative of ( y = 3\tan^{-1}(x+\sqrt{1+x^2}) ) is ( \frac{3}{1 + (x+\sqrt{1+x^2})To find the derivative of ( y = 3 \tan^{-1}(x + \sqrt{1 + x^2}) ), we'll use the chain rule:
[ \frac{d}{dx} \left[ \tan^{-1}(u) \right] = \The derivative of ( y = 3\tan^{-1}(x+\sqrt{1+x^2}) ) is ( \frac{3}{1 + (x+\sqrt{1+x^2})^To find the derivative of ( y = 3 \tan^{-1}(x + \sqrt{1 + x^2}) ), we'll use the chain rule:
[ \frac{d}{dx} \left[ \tan^{-1}(u) \right] = \fracThe derivative of ( y = 3\tan^{-1}(x+\sqrt{1+x^2}) ) is ( \frac{3}{1 + (x+\sqrt{1+x^2})^2}To find the derivative of ( y = 3 \tan^{-1}(x + \sqrt{1 + x^2}) ), we'll use the chain rule:
[ \frac{d}{dx} \left[ \tan^{-1}(u) \right] = \frac{The derivative of ( y = 3\tan^{-1}(x+\sqrt{1+x^2}) ) is ( \frac{3}{1 + (x+\sqrt{1+x^2})^2} \To find the derivative of ( y = 3 \tan^{-1}(x + \sqrt{1 + x^2}) ), we'll use the chain rule:
[ \frac{d}{dx} \left[ \tan^{-1}(u) \right] = \frac{1}{The derivative of ( y = 3\tan^{-1}(x+\sqrt{1+x^2}) ) is ( \frac{3}{1 + (x+\sqrt{1+x^2})^2} ).To find the derivative of ( y = 3 \tan^{-1}(x + \sqrt{1 + x^2}) ), we'll use the chain rule:
[ \frac{d}{dx} \left[ \tan^{-1}(u) \right] = \frac{1}{1The derivative of ( y = 3\tan^{-1}(x+\sqrt{1+x^2}) ) is ( \frac{3}{1 + (x+\sqrt{1+x^2})^2} ).To find the derivative of ( y = 3 \tan^{-1}(x + \sqrt{1 + x^2}) ), we'll use the chain rule:
[ \frac{d}{dx} \left[ \tan^{-1}(u) \right] = \frac{1}{1 +The derivative of ( y = 3\tan^{-1}(x+\sqrt{1+x^2}) ) is ( \frac{3}{1 + (x+\sqrt{1+x^2})^2} ).To find the derivative of ( y = 3 \tan^{-1}(x + \sqrt{1 + x^2}) ), we'll use the chain rule:
[ \frac{d}{dx} \left[ \tan^{-1}(u) \right] = \frac{1}{1 + u^2} \cdot \frac{du}{dx} ]
First, let ( u = x + \sqrt{1 + x^2} ). Then, we find ( \frac{du}{dx} ):
[ \frac{du}{dx} = 1 + \frac{1}{2\sqrt{1 + x^2}} \cdot 2x = 1 + \frac{x}{\sqrt{1 + x^2}} ]
Now, applying the chain rule:
[ \frac{d}{dx} \left[ \tan^{-1}(x + \sqrt{1 + x^2}) \right] = \frac{1}{1 + (x + \sqrt{1 + x^2})^2} \cdot \left(1 + \frac{x}{\sqrt{1 + x^2}}\right) ]
[ = \frac{1}{1 + (x^2 + 2x\sqrt{1 + x^2} + 1 + x^2)} \cdot \left(1 + \frac{x}{\sqrt{1 + x^2}}\right) ]
[ = \frac{1}{2x^2 + 2x\sqrt{1 + x^2} + 2} \cdot \left(1 + \frac{x}{\sqrt{1 + x^2}}\right) ]
[ = \frac{1 + \frac{x}{\sqrt{1 + x^2}}}{2x^2 + 2x\sqrt{1 + x^2} + 2} ]
Finally, multiply by 3 to get the derivative of the original function:
[ \frac{dy}{dx} = 3 \cdot \frac{1 + \frac{x}{\sqrt{1 + x^2}}}{2x^2 + 2x\sqrt{1 + x^2} + 2} ]
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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.
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.
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.
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.
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