differentiate 7 root 3(t)+9 sqrt(t^7)
To differentiate the expression 7√3(t) + 9√(t^7), we will apply the power rule of differentiation and the chain rule.
Let's first differentiate the first term, 7√3(t). Since the square root can be written as a fractional exponent, we have:
d/dt [7√3(t)] = 7 * d/dt [3^(1/2)(t)]
= 7 * (1/2) * 3^(-1/2) * d/dt [t]
= (7/2) * (1/√3) * 1
= (7/2√3)
Now let's differentiate the second term, 9√(t^7). Using the chain rule, we have:
d/dt [9√(t^7)] = 9 * d/dt [(t^7)^(1/2)]
= 9 * (1/2) * (t^7)^(-1/2) * d/dt [t^7]
= (9/2) * (1/√(t^7)) * 7t^6
= (63t^6) / (2√(t^7))
Therefore, the differentiated expression is:
(7/2√3) + (63t^6) / (2√(t^7))
Note that simplifications and further algebraic manipulations could be done depending on the context or requirements of your problem.