Question

attempt 4: the answer is incorrect. 6 attempts remaining. let ( f(x) = 3x^4 ln x ). ( f(x) = (12x)(ln(x)) + left( \frac{1}{x}
ight)(3x^4) ) video example: solving a similar problem

Explanation:

Step1: Identify the function type

The function \( f(x) = 3x^4\ln x \) is a product of two functions, \( u(x)=3x^4 \) and \( v(x)=\ln x \). So we use the product rule for differentiation, which states that if \( y = u(x)v(x) \), then \( y'=u'(x)v(x)+u(x)v'(x) \).

Step2: Differentiate \( u(x) = 3x^4 \)

Using the power rule, if \( u(x)=ax^n \), then \( u'(x)=nax^{n - 1} \). For \( u(x)=3x^4 \), \( a = 3 \) and \( n = 4 \), so \( u'(x)=4\times3x^{4 - 1}=12x^3 \).

Step3: Differentiate \( v(x)=\ln x \)

The derivative of \( \ln x \) with respect to \( x \) is \( v'(x)=\frac{1}{x} \).

Step4: Apply the product rule

Substitute \( u(x) = 3x^4 \), \( u'(x)=12x^3 \), \( v(x)=\ln x \), and \( v'(x)=\frac{1}{x} \) into the product rule formula:
\( f'(x)=u'(x)v(x)+u(x)v'(x)=12x^3\times\ln x+3x^4\times\frac{1}{x} \)

Step5: Simplify the second term

Simplify \( 3x^4\times\frac{1}{x} \). Using the rule \( \frac{x^m}{x^n}=x^{m - n} \), we have \( 3x^{4-1}=3x^3 \). So \( f'(x)=12x^3\ln x + 3x^3 \). We can factor out \( 3x^3 \) to get \( f'(x)=3x^3(4\ln x + 1) \), but the original incorrect derivative had a mistake in the first term's coefficient (it had \( 12x \) instead of \( 12x^3 \)).

Answer:

The correct derivative of \( f(x)=3x^4\ln x \) is \( f'(x)=12x^3\ln x + 3x^3 \) (or \( f'(x)=3x^3(4\ln x + 1) \))