t(x)=((x+4)^0.1-5x)^-5.1

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find derivarive

The given function is t(x) = ((x+4)^0.1 - 5x)^-5.1.

To understand this function, let's break it down step by step:

Step 1: The expression inside the outermost parentheses is (x+4)^0.1 - 5x. This means we need to evaluate the term (x+4)^0.1 and subtract 5x from it.

Step 2: The expression (x+4)^0.1 represents the 0.1 power of (x+4). To calculate this, we need to raise (x+4) to the power of 0.1.

Step 3: The next step is to subtract 5x from the result of step 2.

Step 4: The resulting expression from step 3 is then raised to the power of -5.1.

Let's see an example of evaluating this function for a specific value of x:

Example: Let's say we want to calculate t(2).

Step 1: Start by substituting x = 2 into the expression inside the outermost parentheses: (2+4)^0.1 - 5(2)
=> (6)^0.1 - 10

Step 2: Calculate the 0.1 power of 6:
=> 1.56508458 - 10

Step 3: Subtract 10 from the result of step 2:
=> -8.43491542

Step 4: Finally, raise the result of step 3 to the power of -5.1:
=> (-8.43491542)^-5.1

Thus, t(2) = (-8.43491542)^-5.1.

Note: You can substitute any value you want for x and follow these steps to evaluate the function at that specific value.