Find the derivative of the following function. y=ln(-5x^(5)) Answer: y^(')=

Apply Chain Rule: First, we need to apply the chain rule to differentiate the function y=ln(-5x^(5)). The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.Derivative of ln(u): The outer function is the natural logarithm ln(u), whose derivative with respect to u is 1/u. The inner function is -5x^(5), and we need to find its derivative with respect to x.Derivative of -5x^(5): The derivative of -5x^(5) with respect to x is -5 * 5x^(5-1) = -25x^(4).Apply Chain Rule Again: Now we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us y' = (1/(-5x^(5))) * (-25x^(4)).Simplify Expression: We can simplify the expression by canceling out the x^(4) term in the numerator and denominator, which leaves us with y' = -25 / (-5x).Final Derivative: Further simplification by dividing -25 by -5 gives us y' = 5/x.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the derivative of the following function.

y=ln(-5x^(5))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=\ln \left(-5 x^{5}\right)$ $\newline$ Answer: $y^{\prime}=$

View step-by-step help

Home

Math Problems

Calculus

Find derivatives using logarithmic differentiation

Full solution

Q. Find the derivative of the following function.

\newline

y=\ln \left(-5 x^{5}\right)

\newline

Answer:

y^{\prime}=

Apply Chain Rule: First, we need to apply the chain rule to differentiate the function $y=\ln(-5x^{5})$ . The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Derivative of ln(u): The outer function is the natural logarithm $\ln(u)$ , whose derivative with respect to $u$ is $\frac{1}{u}$ . The inner function is $-5x^{5}$ , and we need to find its derivative with respect to $x$ .
Derivative of $-5x^{5}$ : The derivative of $-5x^{5}$ with respect to $x$ is $-5 \times 5x^{5-1} = -25x^{4}$ .
Apply Chain Rule Again: Now we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us $y' = \frac{1}{(-5x^{5})} \times (-25x^{4})$ .
Simplify Expression: We can simplify the expression by canceling out the $x^{4}$ term in the numerator and denominator, which leaves us with $y' = -\frac{25}{-5x}.$
Final Derivative: Further simplification by dividing $-25$ by $-5$ gives us $y' = \frac{5}{x}$ .