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The chain rule is a powerful technique in calculus used to find the derivative of a composition of functions. It's especially useful when dealing with complex functions where you have one function inside another. The rule states that if you have a composed function, represented as \(h(x) = f(g(x))\), the derivative \(h'(x)\) can be calculated using the derivatives of both \(f(x)\) and \(g(x)\). The chain rule formula is:

• \( h'(x) = f'(g(x)) \cdot g'(x) \)

In the function \(f(x)=10^{\sin x}\), we have a composition where the inner function is trigonometric (\(g(x) = \sin x\)) and the outer function is exponential (\(f(u) = 10^u\)). To apply the chain rule, start by differentiating the outer function, \(10^u\), with respect to \(u\) using the rule for exponential functions. Then, derive the inner function, \(\sin x\), to find \(g'(x)\), which is \(\cos x\). Finally, multiply these derivatives together to get the final result.

Exponential functions are functions of the form \(a^x\), where the base \(a\) is a constant, and \(x\) is the exponent. The base \(e\) is commonly used because it simplifies many calculus operations, but functions with any positive constant base can be handled similarly.The derivative of an exponential function \(a^x\) is given by:

• \( \frac{d}{dx}(a^x) = a^x \cdot \ln(a) \)

In our exercise, we have the exponential function \(10^{\sin x}\), where the base \(10\) is a constant and the exponent is more complex due to the presence of a trigonometric function. To find the derivative, apply the known formula \(\frac{d}{dx}(10^u) = 10^u \cdot \ln(10)\), where \(u = \sin x\). This factor, \(\ln(10)\), accounts for the contribution of the constant base in the derivative.

Trigonometric derivatives refer to the derivatives of trigonometric functions, which are foundational in calculus due to their periodic nature and frequent occurrence in mathematical problems. The basic trigonometric functions include \(\sin(x)\), \(\cos(x)\), \(\tan(x)\), and others. The derivative of \(\sin(x)\) is particularly important and is given by:

• \( \frac{d}{dx}(\sin(x)) = \cos(x) \)

In our function \(f(x) = 10^{\sin x}\), the sine function is embedded within the exponent. By taking the derivative of \(\sin(x)\), we obtain \(\cos(x)\), which represents how the sine function's rate of change is factored into the overall derivative of the composite function. Understanding these trigonometric derivatives enhances your ability to compute and simplify derivatives when they appear as part of more complex expressions.