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How do I find the derivative of #g(x) = 7 sqrt x + e^(6x)ln(x)#?

Answer 1

Ezekiel Alexander

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Answer 2

Axel Alvarez

To find the derivative of ( g(x) = 7 \sqrt{x} + e^{6x}\ln(x) ), use the sum rule and the product rule.

Derivative of ( 7 \sqrt{x} ): ( \frac{d}{dx}(7 \sqrt{x}) = \frac{7}{2\sqrt{x}} )
Derivative of ( e^{6x} \ln(x) ) using the product rule: Let ( u = e^{6x} ) and ( v = \ln(x) ). ( u' = 6e^{6x} ) and ( v' = \frac{1}{x} ). Applying the product rule: ( g'(x) = u'v + uv' = (6e^{6x}) \ln(x) + e^{6x} \cdot \frac{1}{x} )

Therefore, the derivative of ( g(x) ) is ( g'(x) = \frac{7}{2\sqrt{x}} + 6e^{6x}\ln(x) + \frac{e^{6x}}{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.