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(3a^(3)-(4)/(ab))^(4)

66. (3a3−4ab)4 \left(3 a^{3}-\frac{4}{a b}\right)^{4}

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Find derivatives of using multiple formulaeright chevron icon

Full solution

Q. 66. (3a3−4ab)4 \left(3 a^{3}-\frac{4}{a b}\right)^{4}
  1. Identify function: Identify the function to differentiate.\newlineFunction: (3a3−4ab)4(3a^{3} - \frac{4}{ab})^{4}
  2. Apply chain rule: Apply the chain rule for differentiation: ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x). Let u=3a3−4abu = 3a^3 - \frac{4}{ab}. Then, f(u)=u4f(u) = u^4. Differentiate f(u)f(u) with respect to uu: f′(u)=4u3f'(u) = 4u^3.
  3. Differentiate uu: Differentiate u=3a3−4abu = 3a^3 - \frac{4}{ab} with respect to aa.
    u′=dda[3a3]−dda[4ab].u' = \frac{d}{da}[3a^3] - \frac{d}{da}[\frac{4}{ab}].
    First term: dda[3a3]=9a2.\frac{d}{da}[3a^3] = 9a^2.
    Second term: dda[4ab]=4×(−1)×(1ab2)×b=−4ab2.\frac{d}{da}[\frac{4}{ab}] = 4 \times (-1) \times (\frac{1}{ab^2}) \times b = -\frac{4}{ab^2}.
    u′=9a2−4ab2.u' = 9a^2 - \frac{4}{ab^2}.

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