How do you find the derivative of the function using the definition of derivative #g(t) = 7/sqrt(t)#?
The key step is to rationalize a numerator.
I'll assume that you are permitted to use the definition:
(There are other ways of expressing the definition of derivative, but this is a very common one.)
We resume:
Now we can evaluate the limit:
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To find the derivative of the function ( g(t) = \frac{7}{\sqrt{t}} ) using the definition of derivative:

Start with the definition of the derivative: [ g'(t) = \lim_{h \to 0} \frac{g(t + h)  g(t)}{h} ]

Substitute the function ( g(t) = \frac{7}{\sqrt{t}} ) into the definition: [ g'(t) = \lim_{h \to 0} \frac{\frac{7}{\sqrt{t + h}}  \frac{7}{\sqrt{t}}}{h} ]

Rationalize the numerator: [ g'(t) = \lim_{h \to 0} \frac{7\sqrt{t}  7\sqrt{t + h}}{h\sqrt{t}\sqrt{t + h}} ]

Combine like terms in the numerator: [ g'(t) = \lim_{h \to 0} \frac{7(\sqrt{t}  \sqrt{t + h})}{h\sqrt{t}\sqrt{t + h}} ]

Multiply and divide by the conjugate of the numerator: [ g'(t) = \lim_{h \to 0} \frac{7(\sqrt{t}  \sqrt{t + h})}{h(\sqrt{t}  \sqrt{t + h})(\sqrt{t} + \sqrt{t + h})} ]

Simplify the expression: [ g'(t) = \lim_{h \to 0} \frac{7}{h(\sqrt{t} + \sqrt{t + h})} ]

Factor out common terms: [ g'(t) = \lim_{h \to 0} \frac{7}{h} \cdot \frac{1}{\sqrt{t} + \sqrt{t + h}} ]

Take the limit as ( h ) approaches 0: [ g'(t) = \frac{7}{2t\sqrt{t}} ]
Therefore, the derivative of the function ( g(t) = \frac{7}{\sqrt{t}} ) with respect to ( t ) is ( g'(t) = \frac{7}{2t\sqrt{t}} ).
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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.
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