Find the derivative of the following function. y=ln(-2x^(5)) Answer: y^(')=

Identify Function: Identify the function to differentiate. We are given the function y = ln(-2x^(5)). We need to find its derivative with respect to x.Apply Chain Rule: Apply the chain rule for differentiation. 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. In this case, the outer function is ln(u) and the inner function is u = -2x^(5).Differentiate Outer Function: Differentiate the outer function. The derivative of ln(u) with respect to u is 1/u. So, the derivative of ln(-2x^(5)) with respect to -2x^(5) is 1/(-2x^(5)).Differentiate Inner Function: Differentiate the inner function. The derivative of -2x^(5) with respect to x is -10x^(4), using the power rule which states that the derivative of x^n is n*x^(n-1).Multiply Derivatives: Multiply the derivatives from Step 3 and Step 4. We multiply the derivative of the outer function by the derivative of the inner function to get the overall derivative: (1/(-2x^(5)))*(-10x^(4)).Simplify Expression: Simplify the expression. When we multiply (1/(-2x^(5)))*(-10x^(4)), we get -10x^(4)/(-2x^(5)) which simplifies to 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(-2x^(5))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=\ln \left(-2 x^{5}\right)$ $\newline$ 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(-2 x^{5}\right)

\newline

Answer:

y^{\prime}=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $y = \ln(-2x^{5})$ . We need to find its derivative with respect to $x$ .
Apply Chain Rule: Apply the chain rule for differentiation. $\newline$ 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. In this case, the outer function is $\ln(u)$ and the inner function is $u = -2x^{5}$ .
Differentiate Outer Function: Differentiate the outer function. $\newline$ The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$ . So, the derivative of $\ln(-2x^{5})$ with respect to $-2x^{5}$ is $\frac{1}{-2x^{5}}$ .
Differentiate Inner Function: Differentiate the inner function. $\newline$ The derivative of $-2x^{5}$ with respect to $x$ is $-10x^{4}$ , using the power rule which states that the derivative of $x^n$ is $n*x^{(n-1)}$ .
Multiply Derivatives: Multiply the derivatives from Step $3$ and Step $4$ . $\newline$ We multiply the derivative of the outer function by the derivative of the inner function to get the overall derivative: $(\frac{1}{-2x^{5}})\cdot(-10x^{4})$ .
Simplify Expression: Simplify the expression. $\newline$ When we multiply $(\frac{1}{-2x^{5}})\cdot(-10x^{4})$ , we get $-\frac{10x^{4}}{-2x^{5}}$ which simplifies to $\frac{5}{x}$ .