Use derivative formulas to find the derivative of the function.
please show your work
h(x)=10^3-25x^6+3x^15
as written,
h'(x) = 0 - 150x^5 + 45x^2
If 10^3 should really be 10x^3, then modify as needed.
Just simple power rule problem.
does rely on the linearity of the derivative. That is
if h(x) = f(x) + g(x), then
h' = f' + g'
To find the derivative of the function h(x) = 10^3 - 25x^6 + 3x^15, we will use the power rule and the constant multiple rule of differentiation.
Power Rule:
If f(x) = x^n, where n is a constant, then the derivative of f(x), denoted as f'(x), is given by:
f'(x) = n * x^(n-1)
Constant Multiple Rule:
If f(x) = c * g(x), where c is a constant and g(x) is a differentiable function, then the derivative of f(x) is given as:
f'(x) = c * g'(x)
Now, let's find the derivatives of each term in h(x) and combine them.
The derivative of the first term, 10^3, is zero because it is a constant.
The derivative of the second term, -25x^6, can be found using the power rule. The power rule tells us that the derivative of x^n is n * x^(n-1). Applying this to the second term:
d/dx (-25x^6) = -25 * d/dx (x^6)
= -25 * 6 * x^(6-1)
= -150x^5
The derivative of the third term, 3x^15, can also be found using the power rule:
d/dx (3x^15) = 3 * d/dx (x^15)
= 3 * 15 * x^(15-1)
= 45x^14
Putting it all together, the derivative of h(x) = 10^3 - 25x^6 + 3x^15 is:
h'(x) = 0 - 150x^5 + 45x^14
= -150x^5 + 45x^14
Therefore, the derivative of h(x) is -150x^5 + 45x^14.