Differentiate with respect to x.
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Step 1: Simplify and look for algebraic rewrites.
Here the equation is a simplified as we need it, and there are no algebra rewrites. You always have to check though. |
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Step 2: Break the problem down into bitesize chunks based upon the + and – , and identify the derivative rule for each chunk.
Here we have 2 bitesize chunks. – The first chunk will be, 6x , a power rule . – The second chunk, 9 , is a constant .
When looking at the first chunk we must remind ourselves that there is really an unwritten 1 for the power. That is what allows us to apply the power rule . |
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Step 3: Apply derivative rules to each of the chunks you identified in Step 2.
– We apply the power rule recipe, Bring the power down — Subtract 1 from the power, to the first chunk, 6x . The power is 1 . We bring that 1 down in front and then subtract 1 from that power. – We apply the constant recipe, the derivative of a constant is always zero, to the second chunk, 9 .
To simplify remember that .
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Final Result: The derivative of is .
Meaning: – The equation for finding the slope of any tangent line of is . – The instantaneous rate of change for the entire equation is 6 , there is a constant rate of change. Which makes sense when we think about the fact that this equation is a line, and a line always has the same rate of change, the same slope, at every x-value .
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