Is the Derivative the Same as the Slope of a Function?

[Jan Hill]

Homework Statement

is the derivativethe same thing as the slope of the function for which we're finding the derivative?

Homework Equations

The Attempt at a Solution

 

[Mark44]

Homework Statement

is the derivativethe same thing as the slope of the function for which we're finding the derivative?

For a given function (assumed to be differentiable), the derivative is another function that gives the slope of the tangent line at an arbitrary point on the given function.

For example, if f(x) = x², the derivative function is f'(x) = 2x.

If you evaluate f' at say, x = 3, you get the slope of the tangent line to the graph of f(x) = x² at the point (3, 9). Since f'(3) = 2*3 = 6, the slope of the tangent line is 6 at the point (3, 9) on the graph of y = x².

 

[Jan Hill]

chain rule

How do I apply the chain rule to y = x^4sinxcosx in finding the derivative since there are really 3 distinct parts to it.

 

[Mark44]

The chain rule doesn't apply in this problem. Use the product rule.
d/dx(x⁴sin(x)cos(x)) = d/dx(x⁴) * sin(x)cos(x) + x⁴ * d/dx(sin(x)) * cos(x) + x⁴ * sin(x) * d/dx(cos(x))

 

[Jan Hill]

Thank you