Math  /  Calculus

QuestionDifferentiate the function with respect to the independent variable. h(t)=lnt5+4t2h(t)=\begin{array}{l} h(t)=\frac{\ln t}{5+4 t^{2}} \\ h^{\prime}(t)=\square \end{array} \square

Studdy Solution

STEP 1

1. We are given the function h(t)=lnt5+4t2 h(t) = \frac{\ln t}{5 + 4t^2} .
2. We need to differentiate this function with respect to t t .

STEP 2

1. Identify the rule for differentiation to be used.
2. Apply the quotient rule for differentiation.
3. Differentiate the numerator and the denominator separately.
4. Substitute the derivatives back into the quotient rule formula.
5. Simplify the expression to find h(t) h'(t) .

STEP 3

Identify the rule for differentiation to be used. Since h(t) h(t) is a quotient of two functions, we will use the quotient rule:
The quotient rule states that if u(t)=f(t)g(t) u(t) = \frac{f(t)}{g(t)} , then:
u(t)=f(t)g(t)f(t)g(t)[g(t)]2 u'(t) = \frac{f'(t)g(t) - f(t)g'(t)}{[g(t)]^2}

STEP 4

Apply the quotient rule to h(t)=lnt5+4t2 h(t) = \frac{\ln t}{5 + 4t^2} .
Let f(t)=lnt f(t) = \ln t and g(t)=5+4t2 g(t) = 5 + 4t^2 .

STEP 5

Differentiate the numerator f(t)=lnt f(t) = \ln t :
f(t)=1t f'(t) = \frac{1}{t}
Differentiate the denominator g(t)=5+4t2 g(t) = 5 + 4t^2 :
g(t)=8t g'(t) = 8t

STEP 6

Substitute the derivatives back into the quotient rule formula:
h(t)=(1t)(5+4t2)(lnt)(8t)(5+4t2)2 h'(t) = \frac{\left(\frac{1}{t}\right)(5 + 4t^2) - (\ln t)(8t)}{(5 + 4t^2)^2}

STEP 7

Simplify the expression:
h(t)=5t+4t8tlnt(5+4t2)2 h'(t) = \frac{\frac{5}{t} + 4t - 8t \ln t}{(5 + 4t^2)^2}

STEP 8

Combine and simplify the terms in the numerator:
h(t)=5+4t28t2lntt(5+4t2)2 h'(t) = \frac{5 + 4t^2 - 8t^2 \ln t}{t(5 + 4t^2)^2}
The derivative of the function is:
h(t)=5+4t28t2lntt(5+4t2)2 h'(t) = \frac{5 + 4t^2 - 8t^2 \ln t}{t(5 + 4t^2)^2}

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