What is the derivative of 7xy?

Question

Gauge Howard · Accepted Answer

That depends on what variable you want to take the derivative with respect to. With respect to x the answer is

7 \cdot y

, while with respect to y the answer is

7 \cdot x

.
When you take a derivative of a function you get an expression that represents the rate of change or slope of that function. With single variable calculus you usually only ever take derivatives for a function with respect to the independent variable. This is usually represented by x. However, with multivariate calculus you often times have a function defined in three or more dimensions. For instance the above function might look something like this.

z = 7 \cdot x \cdot y

You could then choose to take the derivative with respect to either the x axis or the y axis. If you take the derivative with respect to the y axis you will get an expression representing the rate of change or slope in the y direction, while if you were instead take the derivative with respect to the x axis you would get an expression representing the slope in the x direction. The process to take that derivative is as simple as treating the other variable as if it were a constant like the 7.

\frac{d z}{d y} = 7 \cdot x

\frac{d z}{d x} = 7 \cdot y

Marco Hudson · Accepted Answer

(      7      x      y      )        ′    =    7    x    y    ′    +    7    y  To get this we use the product rule:      d                            (          u          v          )                                      d          x                      =    u    .                  d        v                              d          x                      +    v    .                  d        u                              d          x                    So       d                            (          7          x          y          )                                      d          x                      =    7    x    .                  d        y                    d        x              +    7    y    .                  d        x                    d        x            Which we can write:            (      7      x      y      )        ′    =    7    x    y    ′    +    7    y   or             (      7      x      y      )        ′    =    7          (      x      y      ′      +      y      )

What is the derivative of 7xy?

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