To determine

To find: The differentiation of the function h(t)=6t+16t−1.

Answer

The differentiation of the function h(t)=6t+16t−1 is h′(t)=−12(6t−1)2.

Explanation

Derivative rules:

(1) Quotient Rule: If f(x) and g(x) are both differentiable, then

ddx[f(x)g(x)]=g(x)ddx[f(x)]−f(x)ddx[g(x)][g(x)]2

(2) Power rule: ddx(xn)=nxn−1

(3). Derivative of exponential function: ddx(ex)=ex

Calculation:

The derivative of the function is dhdt, which is obtained as follows,

dhdt=ddt(6t+16t−1)

Substitute 6t+1 for f(x) and 6t−1 for g(x) in the quotient rule (1),

dhdt=(6t−1)ddt(6t+1)−(6t+1)ddt(6t−1)[6t−1]2

Apply the derivative rule (3) and the power rule (2),

dhdt=(6t−1)(6ddt(t)+ddt(1))−(6t+1)(ddt(6t)−ddt(1))[6t−1]2=(6t−1)(6(1)+0)−(6t+1)(6(1)−0)[6t−1]2=(6t−1)6−(6t+1)6[6t−1]2=36t−6−36t−6[6t−1]2

Simplify further as,

36t−6−36t−6[6t−1]2=−12[6t−1]2=−12[6t−1]2

Therefore, the differentiation of the function h(t)=6t+16t−1 is h′(t)=−12(6t−1)2.