f(x)=(-x^(2)+7x-10)/(6x+11)

Identify Function and Rule: Identify the function and the rule to use for differentiation. We have a function of the form u(x)/v(x), where u(x) = -x^2 + 7x - 10 and v(x) = 6x + 11. To find the derivative of a quotient, we use the quotient rule, which states that (u/v)' = (v*u' - u*v') / v^2.Differentiate Numerator: Differentiate the numerator u(x) = -x^2 + 7x - 10. Using the power rule, the derivative of u(x) with respect to x is u'(x) = d/dx(-x^2) + d/dx(7x) - d/dx(10) = -2x + 7.Differentiate Denominator: Differentiate the denominator v(x) = 6x + 11. Using the power rule and the constant rule, the derivative of v(x) with respect to x is v'(x) = d/dx(6x) + d/dx(11) = 6.Apply Quotient Rule: Apply the quotient rule. Now we apply the quotient rule: f'(x) = (v*u' - u*v') / v^2. Substituting the derivatives we found in Steps 2 and 3, we get: f'(x) = ((6x + 11)*(-2x + 7) - (-x^2 + 7x - 10)*6) / (6x + 11)^2.Expand and Simplify: Expand the numerator and simplify. Expanding the numerator, we get: f'(x) = (-12x^2 + 42x + 6x - 77 - (-6x^2 + 42x - 60)) / (6x + 11)^2. Simplifying the numerator, we get: f'(x) = (-12x^2 + 48x - 77 + 6x^2 - 42x + 60) / (6x + 11)^2. f'(x) = (-6x^2 + 6x - 17) / (6x + 11)^2.Check for Simplifications: Check for any possible simplifications. The numerator and the denominator do not have any common factors, so this is the simplified form of the derivative.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$f(x)=\frac{-x^{2}+7 x-10}{6 x+11}$

View step-by-step help

Home

Math Problems

Calculus

Find derivatives of logarithmic functions

Full solution

f(x)=\frac{-x^{2}+7 x-10}{6 x+11}

Identify Function and Rule: Identify the function and the rule to use for differentiation. $\newline$ We have a function of the form $\frac{u(x)}{v(x)}$ , where $u(x) = -x^2 + 7x - 10$ and $v(x) = 6x + 11$ . To find the derivative of a quotient, we use the quotient rule, which states that $(\frac{u}{v})' = \frac{v*u' - u*v'}{v^2}$ .
Differentiate Numerator: Differentiate the numerator $u(x) = -x^2 + 7x - 10$ . Using the power rule, the derivative of $u(x)$ with respect to $x$ is $u'(x) = \frac{d}{dx}(-x^2) + \frac{d}{dx}(7x) - \frac{d}{dx}(10) = -2x + 7$ .
Differentiate Denominator: Differentiate the denominator $v(x) = 6x + 11$ . Using the power rule and the constant rule, the derivative of $v(x)$ with respect to $x$ is $v'(x) = \frac{d}{dx}(6x) + \frac{d}{dx}(11) = 6$ .
Apply Quotient Rule: Apply the quotient rule. $\newline$ Now we apply the quotient rule: $f'(x) = \frac{v*u' - u*v'}{v^2}$ . $\newline$ Substituting the derivatives we found in Steps $2$ and $3$ , we get: $\newline$ $f'(x) = \frac{(6x + 11)(-2x + 7) - (-x^2 + 7x - 10)6}{(6x + 11)^2}$ .
Expand and Simplify: Expand the numerator and simplify. $\newline$ Expanding the numerator, we get: $\newline$ $f'(x) = (-12x^2 + 42x + 6x - 77 - (-6x^2 + 42x - 60)) / (6x + 11)^2.$ $\newline$ Simplifying the numerator, we get: $\newline$ $f'(x) = (-12x^2 + 48x - 77 + 6x^2 - 42x + 60) / (6x + 11)^2.$ $\newline$ $f'(x) = (-6x^2 + 6x - 17) / (6x + 11)^2.$
Check for Simplifications: Check for any possible simplifications. The numerator and the denominator do not have any common factors, so this is the simplified form of the derivative.