why does x^4+7 and x^4 have the same derivative, meaning can someone explain to me why they are both equal to the derivative 4x^3?
The derivative is SLOPE of the curve at x
It does not matter if you move the graph of the function up 7 units, the slope will be the same at each x. Graph it :)
a. Both have the exact same shape of curve, that is, except for position. Derivative means the function which defines the slope at any x, and both functions are the same.
go to your definition of derative.\
Lim d>0 of (f(x+d)-f(x))/d
= Lim d>0 of ((x+d)^4 -x^4)/d
= lim d>0 of (x^4+4x^3*d + ....-x^4)/d
= lim d>0 of (4x^3d + higher terms of d)/d
= 4x^3
Now do the same for f(x)=x^4+7
hint: the lim f(x+d)= (x+d)^4+7
and if you work that it shortly becomes the same as above.
To understand why both functions, f(x) = x^4 + 7 and g(x) = x^4, have the same derivative, we need to have a good understanding of the derivative itself.
The derivative of a function measures the rate at which the function is changing with respect to its input variable, typically denoted as x. It gives us information about the slope of the function at any given point.
In this case, we want to find the derivative of both f(x) = x^4 + 7 and g(x) = x^4. To do that, we need to differentiate each term of the function with respect to x.
For f(x) = x^4 + 7:
The derivative of x^4 is found by applying the power rule, which states that if you have a term x raised to the power of n, the derivative will be n times x^(n-1). In this case, n is 4, so the derivative of x^4 is 4x^(4-1) = 4x^3.
The derivative of a constant (in this case, 7) is zero, as constants do not change with respect to x.
So, the derivative of f(x) = x^4 + 7 is 4x^3 + 0 = 4x^3.
For g(x) = x^4:
Using the same power rule, we differentiate x^4 as 4x^(4-1) = 4x^3. In this case, g(x) does not have any additional constant term, so its derivative is simply 4x^3.
Therefore, both f(x) = x^4 + 7 and g(x) = x^4 have the same derivative, which is 4x^3.
In summary, the reason why both functions have the same derivative is because the derivative of a constant term (such as 7) is zero, and the derivative of x^4 is 4x^3 according to the power rule.