Find the derivative of the function. g(t)=(1)/((7t+1)^(6))

Identify function: Identify the function to differentiate. We are given the function g(t) = (1)/((7t+1)^(6)). We need to find its derivative with respect to t.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 f(u) = u^(-6) and the inner function is u(t) = 7t + 1.Differentiate outer function: Differentiate the outer function with respect to the inner function. The derivative of f(u) = u^(-6) with respect to u is f'(u) = -6u^(-7).Differentiate inner function: Differentiate the inner function with respect to t. The derivative of u(t) = 7t + 1 with respect to t is u'(t) = 7.Apply chain rule: Apply the chain rule by multiplying the derivatives from Step 3 and Step 4. The derivative of g(t) with respect to t is g'(t) = f'(u(t)) * u'(t) = -6(7t + 1)^(-7) * 7.Simplify expression: Simplify the expression. g'(t) = -6 * 7 * (7t + 1)^(-7) = -42 * (7t + 1)^(-7).

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the derivative of the function. $\newline$ $g(t)=\frac{1}{(7t+1)^{6}}$

View step-by-step help

Home

Math Problems

Algebra 2

Sin, cos, and tan of special angles

Full solution

Q. Find the derivative of the function.

\newline

g(t)=\frac{1}{(7t+1)^{6}}

Identify function: Identify the function to differentiate. $\newline$ We are given the function $g(t) = \frac{1}{(7t+1)^{6}}$ . We need to find its derivative with respect to $t$ .
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 $f(u) = u^{-6}$ and the inner function is $u(t) = 7t + 1$ .
Differentiate outer function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $f(u) = u^{-6}$ with respect to $u$ is $f'(u) = -6u^{-7}$ .
Differentiate inner function: Differentiate the inner function with respect to $t$ . The derivative of $u(t) = 7t + 1$ with respect to $t$ is $u'(t) = 7$ .
Apply chain rule: Apply the chain rule by multiplying the derivatives from Step $3$ and Step $4$ . $\newline$ The derivative of $g(t)$ with respect to $t$ is $g'(t) = f'(u(t)) \cdot u'(t) = -6(7t + 1)^{-7} \cdot 7$ .
Simplify expression: Simplify the expression. $\newline$ $g'(t) = -6 \times 7 \times (7t + 1)^{-7} = -42 \times (7t + 1)^{-7}$ .