What is the derivative of #sin 5x#?

1 Answer
mason m
Jan 19, 2016

#5cos5x#

Explanation:

Use the chain rule.

The chain rule states that, in the case of a sine function,

#d/dx[sinu]=cosu*(du)/dx#

More generally, the chain rule says to identify an inside function and an outside function. Here, the outside function is #sinx#, and the inside function is #5x#.

The chain rule then says to differentiate the outside function, and the derivative of #sinx# is #cosx#. With this derivative, plug in the inside function: this gives us #cos5x#.

The final step of this is to multiply the function by the derivative of the inside function, and the derivative of #5x# is #5#.

Thus, the derivative of the whole function is #cos5x*5#, or #5cos5x#.

Using the rule given at the top:

#d/dx[sin5x]=cos5x*d/dx[5x]=cos5x*5=5cos5x#

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