Find the derivatives of f(x) w.r.t. x in the following :
`f(x)=root(3)(ax+b)`

Answer

To find the derivative of the function \( f(x) = \sqrt[3]{ax + b} \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the function in exponent form
The cube root can be expressed using exponents:
\[ f(x) = (ax + b)^{\frac{1}{3}} \]

### Step 2: Apply the chain rule
To differentiate \( f(x) \), we will use the chain rule. The chain rule states that if you have a composite function \( g(h(x)) \), then the derivative is given by:
\[ \frac{d}{dx} g(h(x)) = g'(h(x)) \cdot h'(x) \]
In our case, let \( g(u) = u^{\frac{1}{3}} \) and \( h(x) = ax + b \).

### Step 3: Differentiate the outer function
First, we differentiate \( g(u) \):
\[ g'(u) = \frac{1}{3} u^{-\frac{2}{3}} \]

### Step 4: Differentiate the inner function
Next, we differentiate \( h(x) \):
\[ h'(x) = a \]

### Step 5: Apply the chain rule
Now, we can apply the chain rule:
\[ f'(x) = g'(h(x)) \cdot h'(x) = \frac{1}{3} (ax + b)^{-\frac{2}{3}} \cdot a \]

### Step 6: Simplify the expression
Thus, we can simplify the expression:
\[ f'(x) = \frac{a}{3 (ax + b)^{\frac{2}{3}}} \]

### Final Answer
The derivative of \( f(x) \) with respect to \( x \) is:
\[ f'(x) = \frac{a}{3 (ax + b)^{\frac{2}{3}}} \]

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