How do you find the derivative for #(t^1.7 + 8)/( t^1.4 + 6)#?

Question

How do you find the derivative for #(t^1.7 + 8)/(  t^1.4 + 6)#?

Emilia Aguilar · Accepted Answer

You can use the quotient rule.

You can differentiate a function #f(x)# that can be written as the quotient of two other functions, #g(x)# and #h(x)#, by using the quotient rule #color(blue)(d/dx(f(x)) = (g^'(x) * h(x) - g(x) * h^'(x))/[h(x)]^2#, where #h(x)!=0# In your case, you have #g(t) = t^1.7 + 8# and #h(y) = t^1.4 + 6# This means that the derivative of your function #f(t) = g(t)/(h(t))# will be #f^'(t) = (d/(dt)(t^1.7 + 8) * (t^1.4 + 6) - (t^1.7 + 8) * d/(dt)(t^1.4 + 6))/(t^1.4 + 6)^2# #f^'(t) = (1.7 * t^0.7 * (t^1.4 + 6) - (t^1.7 + 8) * 1.4 * t^0.4)/(t^1.4 + 6)^2# This is equivalent to #f^'(t) = (1/10(17 * t^2.1 + 102 * t^0.7 - 14 * t^2.1 - 112 * t^0.4))/(t^1.4 + 6)^2# #f^'(t) = (3 * t^2.1 + 102 * t^0.7 - 112 * t^0.4)/(10(t^1.4 + 6)^2)# Finally, you can write #f^'(t) = color(green)(1/10 * (t^0.4(3 * t^1.7 + 102 * t^0.3 - 112))/(t^1.4 + 6)^2#

Isaiah Allen · Answer

undefined To find the derivative of the function $ \frac{t^{1.7} + 8}{t^{1.4} + 6} $, you can use the quotient rule, which states that the derivative of $ \frac{f(x)}{g(x)} $ is $ \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} $. Applying this rule, the derivative of the given function is:

$$ \frac{(1.7t^{0.7})(t^{1.4} + 6) - (t^{1.7} + 8)(1.4t^{0.4})}{(t^{1.4} + 6)^2} $$

Emilia Aguilar · Answer

undefined To find the derivative of the given function $\frac{t^{1.7} + 8}{t^{1.4} + 6}$, you can use the quotient rule, which states that if you have a function of the form $\frac{f(x)}{g(x)}$, the derivative is given by:

$$ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2} $$

Apply this rule to find the derivative of the given function. You'll need to find the derivatives of $f(t) = t^{1.7}$ and $g(t) = t^{1.4}$ using the power rule, and then apply the quotient rule.

So, the derivative $ \frac{d}{dt} \left( \frac{t^{1.7} + 8}{t^{1.4} + 6} \right) $ can be computed using the quotient rule.