If y = a^{u(x)}, what is dy/dx?

If y = a^{u(x)}, what is dy/dx?

• A. a^{u(x)} · u'(x)
• B. a^{u(x)-1} · u'(x)
• C. u'(x) · ln(a)
• D. a^{u(x)} ln(a) · u'(x)

Answer: (D)

When the exponent depends on x, differentiate using the chain rule. The derivative of a^t with respect to t is a^t ln a. Let t = u(x). Then by the chain rule, d/dx [a^{u(x)}] = (d/dt a^t |_{t=u(x)}) · u'(x) = a^{u(x)} ln a · u'(x). This requires the base a to be positive (so ln a is defined); if a = 1 the derivative is 0 as expected. So the derivative is a^{u(x)} ln(a) · u'(x).

When the exponent depends on x, differentiate using the chain rule. The derivative of a^t with respect to t is a^t ln a. Let t = u(x). Then by the chain rule, d/dx [a^{u(x)}] = (d/dt a^t |_{t=u(x)}) · u'(x) = a^{u(x)} ln a · u'(x). This requires the base a to be positive (so ln a is defined); if a = 1 the derivative is 0 as expected. So the derivative is a^{u(x)} ln(a) · u'(x).