- #1
lIllIlIIIl
- 7
- 5
Can't figure it out, here's a screenshot with better typography.
Often integral tables can help get an answer but they are acceptable when its a homework assignment. I looked at a few and this integral is not listed. They do show ones where ##ln(x)^n## with n an integer but none with a real number power.I can't imagine where that function came from, but the chain rule is your friend!lIllIlIIIl said:
Your problem is ambiguous, so isn't clear to me.lIllIlIIIl said:
The problem asks for the derivative, not the integral.jedishrfu said:
The derivative of \( (\ln(x))^e \) with respect to \( x \) is \( e (\ln(x))^{e-1} \cdot \frac{1}{x} \). This is found using the chain rule and the power rule.
To apply the chain rule, first recognize that \( (\ln(x))^e \) is a composite function where \( u = \ln(x) \) and the outer function is \( u^e \). The chain rule states that the derivative of a composite function \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \). Here, \( f(u) = u^e \) and \( g(x) = \ln(x) \). Thus, the derivative is \( e (\ln(x))^{e-1} \cdot \frac{1}{x} \).
The natural logarithm function \( \ln(x) \) is involved because it is the base function being raised to the power \( e \). When differentiating \( (\ln(x))^e \), the properties of the natural logarithm and the rules of differentiation for exponential and logarithmic functions are used.
The derivative \( e (\ln(x))^{e-1} \cdot \frac{1}{x} \) is generally considered to be in its simplest form. It clearly shows the application of the chain rule and the power rule. Further simplification is not typically necessary unless additional context or constraints are provided.
Common mistakes include forgetting to apply the chain rule, incorrectly differentiating \( \ln(x) \), and misapplying the power rule. It's important to remember that \( \frac{d}{dx}[\ln(x)] = \frac{1}{x} \) and to correctly handle the exponent \( e \) when applying the chain rule.