Find $\frac { d y } { d x } .$

question 83

Essay

Find $\frac { d y } { d x } .$ (a) $y = 7 ^ { 3 } + x ^ { 3 }$ (b) $y = x ^ { 50 } + 25 x - 1$ (c) $y = \left( x ^ { 3 } - 3 \sqrt [ 3 ] { x } \right) / x$ (d) $y = 2 e ^ { x } + x ^ { e }$

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Definitions:

Causal Relationship

A connection between two variables where a change in one variable (the cause) leads to a change in another variable (the effect).

Standard Error

A statistical measure that describes the accuracy with which a sample distribution represents a population using the standard deviation.

Regression Line

A straight line that best fits the data on a scatter plot, indicating the average relationship between the independent and dependent variables.

Coefficient Of Determination

A statistical measure, often denoted as R-squared, that represents the proportion of the variance in the dependent variable that is predictable from the independent variables, indicating the strength of the relationship.