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Rate of change for y=(-(2)^x) - 1, 2
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Absolutely, I've been sharpening my skills in handling derivative problems, and I'm happy to help. Let's find the rate of change for the expression:
−
dx
d
(−(2)
x
−1)
We can find the rate of change of an expression by taking its derivative. The derivative of a function tells us how fast the function is changing at any given point.
Steps to solve:1. Differentiate the expression term by term:
−(
dx
d
(−2
x
)+
dx
d
(−1))
2. Factor out the constant:
−(−
dx
d
(2
x
)+
dx
d
(−1))
3. Rewrite the exponential term:
−(−
dx
d
(e
ln(2)x
)+
dx
d
(−1))
4. Apply the chain rule:
−(−e
ln(2)x
dx
d
(ln(2)x)+
dx
d
(−1))
5. Rewrite the exponential term:
−(−2
x
dx
d
(ln(2)x)+
dx
d
(−1))
6. Factor out the constant:
−(−2
x
ln(2)
dx
d
x+
dx
d
(−1))
7. Differentiate the variable:
−(−2
x
ln(2)⋅1+
dx
d
(−1))
8. Re-order terms:
−(−2
x
ln(2)+0)
9. Differentiate the constant:
−(−2
x
ln(2)+0)
10. Add zero:
−(−2
x
ln(2))
Answer:
2
x
ln(2)
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