To find the derivative of the function ( y = 3^x ), we apply the chain rule. We start by letting ( u = 3^x ) and thus ( y = f(u) = u ). The chain rule states that:
y′=f′(u)⋅u′Next, we need to determine ( f’(u) ). Since ( f(u) = u ), the derivative is straightforward:
f′(u)=dud(u)=1Now, we compute ( u’ ), which is the derivative of ( u = 3^x ). Using the formula for the derivative of an exponential function, we have:
u′=dxd(3x)=ln(3)⋅3xSubstituting ( f’(u) ) and ( u’ ) back into the chain rule expression gives us:
y′=f′(u)⋅u′=1⋅(ln(3)⋅3x)=ln(3)⋅3xThus, we conclude that the derivative of ( y = 3^x ) is:
y′=3xln(3)This result indicates that the slope of the tangent line to the graph of ( y = 3^x ) at any point ( (x, y) ) is given by ( 3^x \ln(3) ). The function ( y = 3^x ) is a prime example of an exponential function, which increases rapidly as ( x ) becomes larger. The derivative ( y’ ) informs us about the rate of growth of the function at any specific point, while the term ( \ln(3) ) represents the growth rate relative to the base of the exponential function.
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100% |
|---|
Professional Tutors |
| All of our elite tutors are full-time professionals, with at least five years of tuition experience and over 5000 accrued teaching hours in their subject. |
Global |
International Tuition |
| Based in Cambridge, with operations spanning the globe, we can provide our services to support your family anywhere. |
97% |
Independent School Entrance Success |
| Our families consistently gain offers from at least one of their target schools, including Eton, Harrow, Wellington and Wycombe Abbey. |
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