Home > Calculus > Differentiating Logarithmic Functions with Base e

What are the first and second derivatives of #f(x)=-3x^5lnx #?

Answer 1

Cameron Alexander

Here, we'll need the product rule, which states that for #y=f(x)g(x)#, #y'=f'(x)g(x)+f(x)g'(x)#

Solving:

#(df(x))/(dx)=(-15x^4)(lnx)+(-3x^5)(1/x)=-15x^4lnx-3x^4=color(green)(-3x^4(5lnx+1))#

Second derivative follows the same logic:

#(df(x))^2/(dx)=(-12x^3)(5lnx+1)+(-3x^4)(5/x)=-60x^3lnx-12-15x^3#

#(df(x))^2/(dx)=color(blue)(-15x^3(4lnx+1)-12)#

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Answer 2

Christopher Agresta

The first derivative is f'(x) = -15x^4 ln(x) - 3x^4.

The second derivative is f''(x) = -60x^3 ln(x) - 36x^3 - 15x^3 / 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.