How do you differentiate #g(y) =(x^2 + 6) (x^2  2)^(3/2 # using the product rule?
If it takes the form f(x)g(x) as shown above, it becomes f'(x)(g(x) + f(x)g'(x) for the product rule.
In order to calculate the derivative for the chain rule, you must first find the inside derivative and multiply it. This can be done by moving the exponent to the front and subtracting it by 1.
By signing up, you agree to our Terms of Service and Privacy Policy
To differentiate ( g(y) = (x^2 + 6) \cdot (x^2  2)^{\frac{3}{2}} ) using the product rule, follow these steps:

Identify the two functions being multiplied: ( f(x) = x^2 + 6 ) and ( h(x) = (x^2  2)^{\frac{3}{2}} ).

Apply the product rule: [ g'(x) = f'(x)h(x) + f(x)h'(x) ]

Differentiate each function separately: [ f'(x) = 2x ] [ h'(x) = \frac{3}{2}(x^2  2)^{\frac{1}{2}} \cdot 2x ]

Substitute the derivatives and original functions into the product rule formula: [ g'(x) = (2x)(x^2  2)^{\frac{3}{2}} + (x^2 + 6) \cdot \frac{3}{2}(x^2  2)^{\frac{1}{2}} \cdot 2x ]

Simplify the expression as needed.
That's how you differentiate ( g(y) = (x^2 + 6) \cdot (x^2  2)^{\frac{3}{2}} ) using the product rule.
By signing up, you agree to our Terms of Service and Privacy Policy
To differentiate ( g(y) = (x^2 + 6)(x^2  2)^{\frac{3}{2}} ) using the product rule:
Let ( f(x) = x^2 + 6 ) and ( h(x) = (x^2  2)^{\frac{3}{2}} ).
Then, using the product rule, the derivative of ( g(y) ) with respect to ( x ) is:
( g'(x) = f'(x)h(x) + f(x)h'(x) )
Where:
 ( f'(x) ) is the derivative of ( f(x) = x^2 + 6 )
 ( h'(x) ) is the derivative of ( h(x) = (x^2  2)^{\frac{3}{2}} )
The derivative of ( f(x) = x^2 + 6 ) is ( f'(x) = 2x ).
To find the derivative of ( h(x) = (x^2  2)^{\frac{3}{2}} ), we'll use the chain rule and the power rule.
Let ( u = x^2  2 ).
( \frac{dh}{du} = \frac{3}{2}(x^2  2)^{\frac{1}{2}} \cdot 2x = 3x(x^2  2)^{\frac{1}{2}} )
Then, using the chain rule, ( \frac{dh}{dx} = \frac{dh}{du} \cdot \frac{du}{dx} = 3x(x^2  2)^{\frac{1}{2}} \cdot 2x = 6x^2(x^2  2)^{\frac{1}{2}} ).
So, ( g'(x) = (2x)(x^2  2)^{\frac{3}{2}} + (x^2 + 6)(6x^2)(x^2  2)^{\frac{1}{2}} ).
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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.
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7