Power Rule for Derivatives. Example Problem 1

Power Rule for Derivatives. Example Problem 1

f(x)=2x^4 -7x^3 +5x-6sqrtx Find f'(x) 6x^1/2 f'(x)=8x^3 -21x^2 +5-3x^-(1/2) f'(x)=8-21+5-3=-11 Let’s create a function f of x. We’ll say, 2x to the 4th minus 7x cubed plus 5x minus 6 square root of x. Let’s find the derivative off prime, given that f is equal to 1. Now that we know the power rule, we can find f'(x), which is just one of many derivatives.

And by the way, it’s really important that you show labels to everything you do. It says to find f prime of 1 and to do that I’m going to have to find f prime of x. We take the derivative of the exponent by multiplying it with any appropriate constant. Thus, 4 to the third power multiplied by yields 8. x cubed minus 3 times 7 is 21. And x squared plus 1 times 5 equals 5. And then 1 minus 1 equals 0, so there is no x component.

The derivative of 5x is simply 5. And then we can write this as 6x to the 1/2. So let’s subtract 3x to the negative 1/2. So, in one sense, f prime of x is simply 1. Here we 3 have 8 minus 21 plus 5 minus 3. We can add 1 to each of these to get the coefficients. As you can see, 8 minus 21 plus 5 minus 3 gives you a value of negative 11. Thus, the slope of the tangent line at x equals 1 is -11.