Find by implicit differentiation.

Question

Find  by implicit differentiation. ​  ​

Quizplus · Accepted Answer

We will use the chain rule to find the derivative of the given expression with respect to x.

Let y = 11eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11

Taking natural logarithm (ln) of both sides, we get:

ln(y) = ln(11eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11)

Using the properties of ln, we can simplify this expression as:

ln(y) = ln(11) + ln(eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11)

ln(y) = ln(11) + eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11

Now, differentiating both sides with respect to x, we get:

1/y * dy/dx = 0 + d/dx (eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11)

Simplifying this expression and solving for dy/dx, we get:

dy/dx = y * d/dx (eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11)

Now, we need to find d/dx (eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11)

To do this, we will use the chain rule again.

Let u = eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11

Taking natural logarithm (ln) of both sides, we get:

ln(u) = ln(eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11)

ln(u) = eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11

Now, differentiating both sides with respect to x, we get:

1/u * du/dx = eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11

Simplifying this expression and solving for du/dx, we get:

du/dx = u * eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11

Substituting u = eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11, we get:

du/dx = eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11 * eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11

du/dx = eac44f_ed91_62e7_bedf_c32d975aad62_TB7497_11

Therefore, d/dx (eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11) = eac44f_ed91_62e7_bedf_c32d975aad62_TB7497_11

Substituting this value in the equation we derived for dy/dx, we get:

dy/dx = y * eac44f_ed91_62e7_bedf_c32d975aad62_TB7497_11

Substituting y = 11eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11, we get:

dy/dx = 11eac44f_ed91_62e6_bedf_2571a202db63_TB7497_11 * eac44f_ed91_62e7_bedf_c32d975aad62_TB7497_11

dy/dx = 11eac44f_ed91_62e7_bedf_8c3b61e62d0d_TB7497_11

Therefore, the answer is A.

Find by Implicit Differentiation. ​ ​